Edit: Thank you everyone who has told me about the cafeteria at the National Museum of the American Indian. I didn't know about it before. I know about it now.
Don't be fooled by myths of "the first Thanksgiving." Traditionally, white Americans have been far more interested in ignoring or eradicating Native foodways than embracing them. Since this question asks about "Native American" restaurants specifically, I'll focus on a couple of moments in restaurant history where food perspectives of various Native peoples were cut out of wider American fine and family dining. However, an integral part of the story is indeed the targeted destruction of Native food cultures by white Americans - especially since, archaeologists and oral tradition tell us, adoption and modification of foodways rather than wholesale switchover had been the result of intertribal contact before 1500.
The corn/squash/beans agriculture of the Southwest nations had spread east and then north up the Mississippi and tributaries. But the Algonquians and Iroquois, like other Eastern Woodlands nations, would spend the winter in their permanent towns and villages through the spring planting season, then move to waterlines (rivers, lakes, ocean) for a few months. They would catch and preserve fish for the present and for the rest of the year, and return home when the crops needed taking care of and harvest. The foraging and hunting seems to have been traditional to the Eastern Woodlands people before the addition of agriculture--the Algonquians only added farming in the 16th century, it seems, later than the Iroquois. The introduction of new technology and even new foods, in other words, did not have to destroy foodways. Unfortunately, that would become a major reason for the scarcity of Native American or specific Native nations' restaurant cuisine today.
The first stage in American restaurant history I want to look at is the mid-19th century. The rise of "dining establishments" had happened quickly from the age of the smalltime colonial tavern - a skyrocketing American economy drew international businessmen to glittering urban hotels, whose dining rooms became well-known and catered to an audienced used to, above all, French and English food. It was the needs of business that, in America, detached food service places from sleeping, as workers in the earlier 19th century needed lunch on the job.
Why this mattered was the creation of highly "ethnic" communities in cities and rural/township regions through immigration, settlement, and employment patterns. From the 1840s, as midday food service places proliferated and eventually broadened their service, immigrant groups built their own ethnic food places to serve their own communities - the innumerable Biergarten of St. Louis, Missouri, for example. (Budweiser beer was born out of one such - failing! - establishment after the Civil War). For the most part, ethnic restaurants continued to target only their linguistic/cultural community. But it's significant for our overall story because it marks the foundational establishment of ethnic cuisine at a time that explicitly excluded Native nations.
Because, for those of you keeping score at home, that era that America was beginning to host ethnic restaurants was the same period that white America doubled down on its endless campaign of cultural genocide against Indians.
The deportation and exile of Indigenous people to typically isolated reservations - the areas white people didn't want to live! - by logic removed them from the opportunity to plant restaurants in bustling towns and thriving cities. But more importantly, the forced moves inherently ruined existing foodways in a lot of cases (you can't fish for salmon in the richest waters if you don't have access to the richest waters) and white American policies took care of the rest. The kidnapping of Native children and compulsory acquisition of Anglo-American cookery skills at boarding schools and mission schools went a long way towards wiping out traditional patterns.
In some cases, Indian cultures had the ability to adapt traditional foodways to new technology and ideas the same way white Americans were adopting their traditional cuisine. The Objibwa method of drying blueberries was superseded by canning technology and processed sugars, but they nevertheless continued (and continue today) to keep berries as a major part of their diet. (On the other hand, pressure from outside meant the berries were increasingly used in very European pies and European-style puddings, not ones thickened by corn as had been prepared for centuries.) In other cases, though, traditional knowledge was lost - actively destroyed - altogether.
Back on the restaurant scene: American xenophobia and (ironically) nativism in the early 19th century kept the ethnic restaurants and ethnic food - above all the "spicy" dishes of Italians and Chinese and other not-white-enough immigrants - somewhat sequestered in their own communities. However, one should note that the increasingly shrill anti-flavor voices were probably reacting to an increased broader public acceptance of said edible food. Some of the early cracks in the shell came through the World's Fairs of the late 19th and early 20th centuries, where Americans could dine on the "cuisine of foreign countries". Of course, this meant largely foreign white countries. The World's Fairs were rather better known for showing off the technological and cultural achievements of one set of countries and peoples, and showing off individual, living human beings of another set of countries and peoples as curiosities in and of themselves. So again, Native Americans - and here we should note also the scarcity of sub-Saharan nations' ethnic restaurants in the U.S. - were left out of the broadening of the white American palate.
The explosion in cookbooks and preserved foodstuffs in the early/mid-nineteenth century was actually a time Native women, in particular, tried to capture and assert the strength of their cultural foodways. In 1933, the Indian Women's Club of Tulsa, Oklahoma produced the first Native cookbook. You can read one recipe, from Creek Indian and club president Lilah Lindsay, on Google Books! Amanda Amanda's Choctaw Indian Dishes in 1935 seems to have been the first professional cookbook promoting one tribe's foodways.
Native women's efforts in the cookbook scene went further towards preserving various cultures' foodways for themselves and for each other, than promoting it to a broader American public with more money and more desire to "eat out" for entertainment and family socializing. The 1940s seem to have been the era when "dining ethnic" took off as a popular social act. Especially in major cities, immigrants began to establish restaurants outside their country-of-origin neighborhood.
However, once again, the foodways that became popular with white people were most often associated with Europe - in particular France, at this time, actually. (AskHistorians has some experts on Asian-American food cultures who will know more about that particular evolution). And once again, Native Americans were systematically excluded from this development, with the endurance of the reservation system and the difficulties or impossibilities of living an explicitly and marked Native life as a community outside.
Native nations were also excluded from the rise of "American regional" dining, which is generally considered to have been born with the Four Seasons in 1959. "Regional" did not include the Native map of America.
Every history of Native foodways will end with a spot of hope for the future. The tradition of cookbooks established by Lilah Lindsay and the Indian Women's Club of Tulsa, to which we probably owe a sizeable debt for preservation of traditional recipes through the mid/late-20th century rise of TV dinners and standardized fast food fare, flourishes; maybe you have had the chance to be a guest at a powwow or attend a restaurant featuring traditional local Native foods as opposed to simply appropriating the name.
I hope that a lot of you really interested in the question of why most of us don't have much exposure to Native American foods today will head over to /r/IndianCountry and read or ask a little about how to get that exposure. :)
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Some further reading, as requested (thank you for your patience):
To add on this if I'm allowed... what is an example of native American food. I know it likely varied by region. We all know the stereotypes of plain dwellers eating buffalo and the early settlers learning how to grow corn from the local natives. But does anyone know if there is any native cuisine?
Founder/mod of /r/IndianCountry reporting in.
The questions as parsed by OP and labeled by this mod team focus on the modern period, which is often neglected when discussing Native Americans. We've aggregated a few stories about this found by searching for "food" or "cooking" or "chef." You'll likely find a few examples of Native American restaurants offering interpretations of specific regional cuisine among the searches and other submissions here you may see.
#Mainstream Examples Hidden in Plain Sight
The glib answer to both questions is "We call it Famous Dave's, given that its founder, David W. Anderson is Choctaw and Ojibiwe, formerly AS-IA of DOI (2004-5).
Going more local as to Washington, DC, there's the Mitsitam Cafe at the National Museum of the American Indian. It's the most ostensibly Native American restaurant in the city, offering a variety of cuisine. The fry bread was a problem until Executive Chef Freddie Bitsoie (Diné [Navajo]) began his tenure.
Going deeper, people often do not consider the deep Native American roots in Soul Food and Southern Cooking generally, which borrows staples and dishes from Southeastern Tribal Nations.
#A Local Example Hidden in Plain Sight
This will be as granular I can make it and I hope it doesn't result in loss of appetite.
My Tribal Community is Pamunkey, known as "Pocahontas' people" by mainstream references, recently acknowledged by the US in 2016. Our most recent treaty is the 1677 Treaty of Middle Plantation (modern Williamsburg) and the Pamunkey Reservation was established by treaty, prior to 1658. However, despite my family maintaining their ties to Pamunkey, mine is largely an Urban Indian experience in the DMV (DC/MD/VA region), shaped by the Jim Crow South, segregation, and the post-segregation periods. Save for a few families who remained exclusively on either Pamunkey or Mattaponi Reservations, the vast majority of the regional "Indian Experience" has been a combination of rural and urban for the past 200 years.
Some of us have opened restaurants and many of us have attempted to blend in by not being explicit about race; we are deeply in the minority and, frankly, it's often a political and economic liability to declare indigeneity in this long, insistently racially binary region. It's not entirely by free personal choice, as the Racial Integrity Act of 1924 criminalized off-reservation Indian identity in Virginia until Loving v. Virginia and regional state-sponsored document genocide efforts continued afterwards, arguably to this day in weaker forms. Long contact means varying degrees of mixture with other communities. Racial purity is a unrealistic and unhealthy luxury that we could not and cannot afford, but local Indians have a good idea of what other local Indian families look like, our surnames, our hometowns, and our ancestral holdings. Some of us don't have the option of passing for White or Black and sometimes we personally pay a kind of "blood price" for the the Tribal Communities we are born into, be it at the altar, on the street, or whether our employer has a strong opinion about local Indians (as opposed to people from non-local Tribes).
The local Native American scene in the DMV is split by State, Tribal Nation, Federal Acknowledgment status, Family, Employer, and Organization; it's rather segregated, despite areas of overlap. Southern Maryland had a particularly active local indigenous community along Route 301, which included a nightlife that the older generation might tell you about when their wives aren't in earshot.
This is a long-winded way of saying that there is a non-zero chance of eating at a Native American restaurant in the DMV without knowing it.
The "World Famous" Florida Avenue Grill, which bills itself as the "Oldest Soul Food Restaurant in the World" is undeniably a historical and popular establishment, and also arguably the best known Native American restaurant in DC that you've never heard or thought of as such.
Its founders, the Wilson family, are Occaneechi (now recognized by North Carolina as the Occaneechi Band of Saponi) and I've never heard Lacey C. Wilson, Jr. describe himself in any other way. He's bled for that, but you wouldn't know it unless you were one of the local families. My mother conducted interviews with him, but she has exclusive control of the material, which will hopefully not pass into my possession any time soon. The food is fantastic, although I haven't eaten there in years.
I previously worked on cultural resources (gathering historic research, identifying culturally-sensitive plants and natural resources, and communicating with local tribal representatives) for military installations in the Pacific Northwest. Just moved to the Southeast where I helped with an intertribal powwow, but I'm still learning about the local history and tribes here. When I use the word "American" or "Native/Native American" I refer to North American or North American Indigenous, respectively, though again my expertise is more from the PNW.
Much of what is currently, regularly eaten by many Native Americans is not the same as what their ancestors traditionally ate and prepared. Communities on some reservations sometimes face the same sort of food deserts that many non-Native American neighborhoods suffer from--cheap junk food and limited fresh produce. Access to traditional ingredients or food knowledge can be very limited. For example, on a lot of the Army training grounds in WA, the military is supposed to immediately cease training to allow (some, depending) tribes to forage for native plants needed for traditional food preparations and ceremonies, though I've personally never met anyone who said they recall them actually exercising that right. These plants don't exactly grow on the most accessible terrain. And when I was gathering information on native plants for signage, I discovered that much of it had traditionally been eaten--but were often regarded as famine foods, or had challenging preparations (such as Chinook acorns, which required curing in urine). So it would be difficult to open a commercially viable restaurant based on either current Native eating habits or using some of the more hard-to-access or prepare foods.
We do, however, eat and prepare plenty of foods that stem from traditional Native American cuisine in much of what we regard as wider "American" cuisine. The use of green chili in the Southwest, cornbread/corn pone (the word pone itself comes from Eastern Algonquian languages), succotash, the use of filé powder in gumbo, and of course smoked salmon and jerkey are examples. The pit-style form of barbeque is believed to derive from a traditional style of cooking meat observed in the Amerindians from the Caribbean up to as far as New York. So you could say that restaurants serving uniquely American fare already have some native cuisine incorporated into them. As far as whether a Native American restaurant "was ever popular in the US" in the past, prior to say the 2000s, this might be the closest you'll get.
There is definitely growing interest in reviving Native cuisine in the PNW, Canada, and Mid-West US, I believe in part due to increased curriculum focusing on native plants and food preparation at some of the local colleges (like Evergreen State) and Northwest Indian College. Indian Colleges especially seem to be finding creative ways to connect students with their heritage such as in their food science and culinary arts departments. Sources for what they teach include documents like The Ethnobotany of Western Washington, in which the food and cultural values of various native plants were recorded for various PNW tribes (interestingly the interviewer tried to discuss each plant with both a man and woman from each tribe, since women were often more familiar with the culinary and medicinal uses and men tended to be more familiar with the plant's value for nets or woodworking), and Buffalo Bird Woman's Garden, which is a direct recounting of a Hidatsa Indian woman's descriptions of her village life, food preparation, and agriculture at Like-a-fishhook village prior to 1885. Lots of descriptions of food in that, and it's now being used by chefs such as Sean Sherman to recreate native cuisine. His consulting company, The Souix Chef, has been working on this sort of research and promoting healthy native cuisine. Ever since around 2010 there has been increased attempts at Native American restaurants, though because the genre of "Native" fine cuisine is still attempting to define itself, it's been challenging for some of the restaurants. Landlords know that it's not exactly a Yelp category that most customers will be searching for, so business loans and securing space can be more challenging than when starting restaurants of other cuisines. There are debates as to how closely Native American cuisine should stick to its historic roots. And items such as the frybread at the National Museum has been controversial due to its origins as an essentially non-nutritious survival staple on reservations--a last resort when they were denied access to traditional ingredients, and a reminder of the injustice. And different tribes, and sometimes each family, has their own way of making and serving it, so it's difficult to say whose recipe (if any) should be a representative of a whole continent's traditional food, especially since that one Native restaurant might be the only direct experience with Native food that most folks might experience. Still, there's increasing research and education on traditional Native American cuisine, as well as increasing public awareness (somewhat, at least in some places) and demand for more unique restaurants, so there's all the factors that could hopefully lead to a flourishing new restaurant category.
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Edit: Seriously, stop just posting the menu for the NMAI cafeteria. We're handing out temp bans for people who do that. (2) To answer what someone asked me via PM, yes, it can be part of an answer if you want to provide an easily accessible illustration, but it needs to be a) contextualized and b) presented in a way that answers the actual question, as a single restaurant in a museum about that very culture doesn't speak to popularity, or larger trends in Native American cuisine vis-a-vis the general public. Additionally, we would expect to see at least some explanation of how the food served there compares to the originals that they represent.
As an add-on question, how were things such as vegetables prepared among natives? Did they fry or bake anything? Were the culinary talented recognized for their 'dishes' like chefs in the western world are?
As a follow up question, what about cookbooks? I know that cookbooks from the middle ages Europe and early America are studied and occasionally the recipes are made, but I've never heard of Native American cook books. Do any exist? Was there a time in US history where they became popular? And what common foods today are influenced by Native American foods and techniques?
What an interesting question, thanks OP.
Edit: I asked my wife about this and I posted her answer in another top comment.