You're not from New England, are you. There hasn't been very much immigration to the US directly from France, at least not in comparison with other European countries like Poland or Ireland. There was, however, a lot of immigration to certain parts of the US during the late 18th and early 19th centuries from Quebec. So yeah, French-Americans are a thing, it's just that they're more like French-Canadian-Americans.

They certainly do exist. The "D" in FDR is for the Delano family, a French-American family from which he was maternally descended. Plenty of other distinguished Americans have had French ancestry, such as Jacqueline Kennedy (née Bouvier), Éleuthère Irénée du Pont, and Jack Kerouac. France ruled a vast swathe of the modern United States until the Louisiana purchase of 1803. They left behind plenty of place names, from Boise, Idaho to St. Louis, Missouri. Louisiana Creole French is still spoken today, along with Canadian French spoken among more recent arrivals in New England.

There are millions of French-Americans, but they are still less than 5% of the population. If your question is why there are not more of them, then someone more qualified than me will need to explain the factors as to why the French were seemingly less likely to emigrate than eg. the Germans or Irish.

There are relatively large communities of French-Americans living in the United States, you probably just know them by their more common name: cajuns/acadians.

I can speak for the French-Catholic shame and subsequent assimilation experience in Maine. The French-Catholic population were often humiliated by their Anglo-Protestant counterparts, with slights as small as being forced to sit in the back of class rooms, being forced to sit for the second part of the Lord's Prayer (again in classrooms) and sometimes being violently persecuted. Being uneducated and being resigned to mill work, many began the quick process of assimilation: anglicizing names, speaking English, so much of the French culture died out quickly in these towns.

I can speak for my Grandfather, who was born in St. Joseph de Beauce, and moved to Central Maine when he was young. He spoke of facets of the KKK cross burning on French Canadian's lawns. Eventually the mill workers and loggers (mostly French Canadian immigrants) got angry enough and fought back against the KKK, but the persistent shame of being primarily Francophone and catholic (mostly brought in public school, of all places) forced him to forgo his heritage. All of my great-uncles and great-aunts spoke French, but if I entered the room, all would switch to English very quickly. I asked my grandfather to speak to me in French, and he gave me a very gruff, "No need for that, you're in America." The only time we heard him speak French was when he was particularly angry.

Here is an interesting essay/interview on the topic of French Catholic persecution in Maine: http://www.fawi.net/ezine/vol3no2/Lord.html (sources in the essay)

The author speaks on three factors that lead to a lot of French heritage being lost: 1.) The Corporation Sole Controversy. 2.)The English only bill passed in the 1920's. 3.)The New Deal National Recovery Act caused by the Great Depression.

Before the Seven Years War, what is now northern coastal Maine was inhabited by the Acadians, that is, French immigrants living along the northern coast of North America (hence, Acadia National Park which is right outside Bar Harbor, Maine). As a result of the Seven Years War / French Indian War, these Acadians were more or less banished by the British victors. The Acadians were sent to French/Spanish New Orleans and other parts of coastal Louisiana where they gradually became "Cajuns". The process of transferring the Acadians to the Gulf of Mexico was very harsh, many Acadians died as a result of being dumped off in an unfamiliar, marshy, mosquito-ridden landscape. Adding to the death rate was disease, unfamiliar soil, and general tension with the British (it’s said that 1/3 to more than half the Acadians died on the ships that brought them to Louisiana). The British actually recently apologized to France, and the Cajuns in particular, for this banishment. Some of these would-be Cajuns ended up living in the Barataria region of southern Louisiana (bayou country)- a region that became notorious for pirates (Pirate Jean Lafitte).

edit: grammer

I know a lot of French families like those in my ancestry have anglicized last names. People of French ancestry have been here for so long that they have blended generically into the English/Scotch Irish/Native American/French/Other Western European "American" mixture. Those other groups you list are second wave immigrants which maintained their group identity longer.

Is this really a history question? I think it's flawed too. It assume that people actually used those terms and most of us rarely if ever hear people referred to as those things. Those terms were only invented in recent times to satisfy the PC crowd and I don't think you'll be able to find much support that those terms were used in history.