That paleo-Indians arrived in the new world by way of Beringia is now settled science. I recommend this post by /u/RioAbajo with further commentary by /u/Reedstilt. There is archaeological, linguistic and genetic evidence supporting that Native Americans are related to and likely descended from populations in central Siberia and entered the New World via Beringia and then followed an "ice free corridor" on the east flank of the Rockies (unlikely) or proceeded down the coast via boat or coastal route (most likely).

Dr. Keene is an accomplished specialist in Native American Studies and appears to be arguing from a position that the origin stories of Native American groups should provide explanations of native origins in North America. Dr. Keene is a Native American and has written extensively on cultural appropriation. She has an Ed.D. from Harvard, but does not appear to have made any academic contributions to the question of the origins of Native Americans and routes to the New World.

There are two published theories of how the new world was occupied. The Beringia hypothesis has been around since the 1930s and was promoted in early studies by Paul S. Martin, Alex D. Krieger and others. Modern proponents of the Beringia land bridge theory include a number of prominent archaeologists, like Don Grayson, Tom Dillehay, Jim Adovasio, Gary Haynes and many others. Perhaps the most prolific current writer on the subject is David Meltzer. There was another very short lived theory on paleo-Indian migration that suggested that early humans in the New World got here via a land bridge or by boats from Europe and that the famous Clovis technology had its roots in the Solutrean lithic tradition. This theory, proposed by Dennis Stanford of the Smithsonian, has been largely discarded by now. As Meltzer puts it:

If Solutrean boat people washed up on our shores, they suffered cultural amnesia, genetic amnesia, dental amnesia, linguistic amnesia and skeletal amnesia. Basically, all of the signals are pointing to Asia as the origin of the first Americans.

So my take on the question is that the Beringian Landbridge theory is still the dominant paradigm in the discipline. I will get some references for you in a minute.

Addendum: Dr. Keene's assertion is based on a 6 part series of articles, that can be found here that was originally published in "Indian Country" by historian Alexander Ewen. Ewen's approach (and this is necessarily a simplification) appears to be that the level of debate over facts about original colonization of the new world found in the archaeological literature, like controversies over dates and dating techniques, disputes over an ice free corridor, the Solutrean hypothesis and arguments over genetics and linguistics render the entire debate pseudoscientific. This is putatively supported by "evidence" Ewen provides that American archaeology has its roots firmly embedded in a colonial foundation that suffers from an approach that has relied on social Darwinism and eugenics as explanatory vehicles in the past. While many of Ewen's assertions are worthy of careful review, his conclusion that the body of study on Paleo-indian movement into the New World is pseudoscience just cannot be supported. Further, his assertions that the historical level of debate over subjects like "Clovis first", "Ice-free corridor" and very early dates is evidence of illegitimate science is nonsense. The disputes are, in fact, evidence that positions and arguments have been and continue to be rigorously evaluated.

Sources:

Fitzhugh, Drs. William; Goddard, Ives; Ousley, Steve; Owsley, Doug; Stanford, Dennis. "Paleoamerican". Smithsonian Institution Anthropology Outreach Office.

The peopling of the Americas: Genetic ancestry influences health. Scientific American.

Strangers in a New Land: What Archaeology Reveals About the First Americans by J. M. Adovasio, David Pedler

First Peoples in a New World: Colonizing Ice Age America by David J. Meltzer (2009)

I'm no historian but a biologist, and I have a hard time understanding where the tweets you link to are coming from.

The current consensus is very much that the first humans to the Americas came from Eastern Siberia through what is today the Bering Strait. This has been pretty conclusively demonstrated not only through archaeology but from genetics.

However, what is still up for discussion is when, how many times, and through which route these humans entered the Americas. The current mainstream hypothesis is that it happened over the dry land of Beringia which connected Siberia and Alaska when the sea level was lower during the last Ice Age. The competing hypothesis is that they were a coastal civilization that followed the coast in boats, maybe similar to modern Inuits. If more than one group arrived, both hypotheses may indeed be right. See this recent blog post for a summary with further link.

The timing of the migration is also hotly debated, as the main archaeological material suggests the Siberians arriving not much more than 12500 years ago. However, there are some (controversial) sites that may suggest an even older immigration event, which may not necessarily have left modern genetic traces.

Different immigration events may of course have followed different routes. But they all have in common that they show immigration through the Bering region. There is one competing hypothesis, the so-called soultrean hypothesis, stating that at least one wave of immigrants came from Western Europe, but this is highly controversial and seems to have lost ground as the advent of ancient-DNA analysis have found no support for it.

Why it would be fashionable to deny immigration from Siberia, I have no idea about, but I would be interested in theories? What is the alternative they prefer?

Here is a popular science article referring to a relevant recent DNA study from the Univ. of Kansas

Here an article with a little background and links to other recent studies

I'm going to go a bit against the grain here and say that yes, the popularity of the Bering Strait theory is losing steam. Since 1997, with the acceptance of an archeological site in Monte Verde, Chile, that dated human occupation back to 14,800 before present (BP), which was a full millennia earlier than it was previously thought humans had been in the Americas, there has been growing debate. A nice overview of it can be found at the National Parks Service website, which I will link here: NPS Bering Strait and alternatives

Lets dive into the evidence though, and a bit of the history here. We can begin with Monte Verde. Thomas Dillehay has a great work on this titled The Settlement of America, a New Prehistory. What makes Monte Verde special is that it is widely accepted as an archeological site that does definitely showcase that humans were here before the Last Glacial Maximum.

So Monte Verde was discovered in 1975, and in 1977, Dillehay began excavating there. In 1982, radiocarbon dating of items in the site was done, and it was discovered that the site in general went back 14,800 years. It would take another decade and a half though for wide acceptance of this dating, and that occurred in 1997, when 12 other archeologists revisited the site.

So why is this important to this discussion? Because various studies have shown that the Monte Verde site is too early for it to have been occupied by individuals who migrated via the Bering Strait. One of these studies, from 2016, titled Postglacial Viability and Colonization in North America's Ice-Free Corridor, published in the journal Nature, argues that the Bering Strait route only became a viable path around 12,600 years ago, when the first plants and animals started showing up in the area. The authors of this article go on to say that it is unlikely that the earliest migration of humans to the Americas used this route, but that later groups may have used the path.

Just a note on the Monte Verde site. In 2015, an article titled New Archeological Evidence for an Early Human Presence at Monte Verde, Chile, was published in Plos One (it can be viewed here. Dillehay was one of the archeologists who went back to Monte Verde and helped author this study. What they discovered is that the site may go back 18,500 years. Interestingly though is that they also discovered the presence of items at the site that aren't found their naturally. Their argument is that those items were brought to Monte Verde from other people who brought them (or some of them) from far away.

Another take away from that article is the landscape of the debate now, that the authors point out in their introduction. They make the claim that the consensus among experts is that humans did arrive in the Americas at least 15,000 years ago. And while it is accepted that some humans would eventually travel to the Americas across the Bering Strait, the old model (the Clovis-first model, that said that the first humans came here via the Bering Strait) "no longer explains the peopling of the New World." And this all has led to new debates, such as, how many migrations were there? Did the first humans arrive here by land or the coastline?

There are other issues though too. In 2010, an article titled Clovis and Western Stemmed: Population Migration and the Meeting of Two Technologies in the Intermountain West was published in American Antiquity. The authors were Charlotte Beck and George T. Jones. One thing that they point out is that the chronology of the Clovis-first model has broken down in recent years. For instance, in 2003, it was discovered that the Ushki site, which had long been held as the earliest site in Western Beringia, which also was seen as a prime candidate for a Clovis ancestor, wasn't as old as thought; it only dated back 11,000 years. This has caused a dilemma now as it erased the progenitor in Siberia, and has made the Clovis origins more complicated.

Another major issue is with the distribution of items associated with the Clovis people. For instance, the fluted point that we know as a distinctive Clovis technology may have actually originated in the east. As Beck and Jones point out though, while that may be the case (based on the distribution density of such items), we cannot be certain until more research is done.

One of the arguments that the authors make though is that the distribution of these items, that seem to spread across the continent very rapidly is that the Clovis people weren't migrating to these new areas, but that they were trading technology with other groups. The key thing here is that it is expected that other groups were here before the Clovis people. They actually emphasize that point, that people were at least in the Intermountain West before the Clovis people were present anywhere.

Their suggestion, and a view that they say is growing more probable, is a Pacific coastal route for early North American entry.

Now, these authors aren't claiming to be drawing the entire picture. A main point that they are trying to get across is that the migration process is much more complex that we imagined. We have to consider that it may be that people migrated here in a variety of different ways, and entered the continent at different locations.

So to sum up, the Bering Strait theory is losing steam. While it is certain that some people migrated to the Americas through the Bering Strait, it now appears that even before that, people were living in the Americas. Most likely, there were different migrations events into the continents, and the routes were varied as well. However, some of the things Dr. Keene is saying (or more specifically linking to, such as the 6 part article that debunks the Bering Strait theory) aren't correct either. As I believe this post shows, for instance, the Bering Straight theory isn't some unshakable idea within scholarship. While it isn't fully rejected, its accepted that we have to rethink it and acknowledge that it was only part of the migration.

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