The same story has been told many times:

During the 1950s, the development of radiometric dating technologies helped renew interest among archaeologists in studying processes of migration and change. Such analytical technologies helped researchers exploit the full range of evidence found at sites, allowing them to make claims about the absolute dates of organic artifacts, the origins of raw materials, and the ancient climatic and geological setting. An explosion of claims about the "earliest Americans" followed, but by the mid-'80s the hype had fizzled, and attentions turned to some specific sites of interest. Their proponents fought an uphill battle and risked their careers, facing a "toxic" environment of laughs and sneers when they presented at conferences. But in 1997, Tom Dillehay invited a team of supporters and skeptics to Monte Verde in Chile, a site he had excavated since 1977. Dillehay had published some just slightly pre-Clovis dates from the site 15-years prior; unlike many other sites, these were associated with a diversity of signs of human occupation that had been fortuitously preserved in an anaerobic peat bog. After the visit, the consensus was unanimous: Monte Verde predated the Clovis culture. The resultant article in American Antiquity "broke the logjam" of pre-Clovis skepticism and was the archaeological equivalent of "breaking the sound barrier." From this point on, those who continue to question pre-Clovis dates do so out of dogmatic ignorance.

But that's the pop science narrative version, and, since you're asking here, I think you know that there's more to the story. Why is this the question that people get hung up on?

Another user has already covered the technical portions of this question. The Clovis people post-date the Last Glacial Maximum, and therefore had an obvious route for migrating into North America, they aligned with the extinction of many megafauna species, and there was little to no evidence of anything beneath Clovis settlements. These are rather indirect arguments, but they've been important enough to some people that it's elevated discussion beyond a simple "who was first."

I'd like to provide some historical perspective on this question and an epilogue on the lessons we should learn from it.

Answering these sorts of questions is difficult because we can't get inside critics' heads and measuring "consensus" is a logistical and theoretical nightmare. It requires engaging not with published, peer-reviewed articles, but with the ephemera of academia: the conversations at university-hosted conference parties, the e-mails after guided site tours, and the comments of whoever happens to be deciding grants at a given institution in a given year. Usually I would put a wry, denigrating disclaimer here about how my perspective is informed by "my Facebook feed which is full of archaeologists" and "years of moderating and participating in pertinent Reddit forums," but I do think answering your question in a meaningful way requires beginning with that material as a legitimate source.

A quick look at these sources tells us that Clovis-first is a long dead theory. I do not recall encountering any controversy about Clovis-first as an undergraduate; comments from /r/archaeology suggest that this applies similarly to those of us in the community who started our archaeological education in the '90s. A 2011 survey published in the newsletter of Society for American Archaeology showed only 10 respondents out of 150 who disagreed the Monte Verde represented a sound pre-Clovis occupation. The discovery of ancient footprints at White Sands National Park was met with enthusiasm in my social media circles. The textbook I last used for teaching Intro to Archaeology, the 2013 7th edition of Images of the Past, starts its section on the peopling of the Americas with "The discoverers of North America walked to their new homeland sometime before 15,000 years ago" and cites 8 different pre-Clovis sites.

Let's instead, then, focus on the specifics that lead to the oh-so-important 1997 Monte Verde visit.

The great antiquity of Clovis-style projectile points was first demonstrated by their close association with now extinct megafauna. By the 1920s, the "Clovis culture" was recognized as the earliest culture of the Americas, and for many decades this was the most reasonable argument. No other diagnostic (i.e. stylistically identifiable to a certain culture) artifacts were so closely connected to mammoths and other predators, Clovis points were found widely distributed across North America but retained a significant degree of similarity, and no cultural deposits had been found beneath Clovis sites. Clovis First therefore predates our ability to actually date Clovis artifacts.

The introduction of radiometric dating entirely altered what it meant to claim something was "first." Previously, chronological claims were based primarily on stratigraphy or artifact seriation, i.e. ordering objects based on incremental changes in technology or decoration. These are fine methods, but require your excavations to yield a certain number and quality of diagnostic artifacts. As such, the first chronologies of early Americans were based on typologies of arrowheads and other projectile points that could be used to associate a given site with a known culture. Clovis wasn't "first" because it was the "oldest," it was first "by default:" nothing else was older. Proving something was pre Clovis implicitly entailed proving it was not Clovis.

Radiometric dating meant that "the oldest site" was now something that could be absolutely measured, with no need to demonstrate the site's relative association to another established first.

Yet the proliferation of claims about pre-Clovis sites that began in the 1960s wasn't simply about archaeologists playing with their shiny new presents from nuclear Santa Claus, but about testing an archaic paradigm of placing sites in a timeline. It's not worth our time to dig into these studies, because the technology was still in its infancy, and many of the sites have since fallen off the radar. What's most interesting to note is just how many studies were published during this period that gave pre-Clovis dates but didn't seem particularly interested in that whole debate. Indeed, neither Dillehay's initial 1982 publication of the Monte Verde dates in Journal of Field Archaeology, his 1984 Scientific American article, nor his 1988 Nature article mention "Clovis."

There are a number of good resources relating to early settlement of the new world in the FAQs for this sub, I recommend this post by u/RioAbajo. I think it fairly clearly lays out that beginning in the mid 20th century, with the discovery of Clovis sites and the advent of C14 dating a model of the New World diaspora was adopted. Because no earlier sites than Clovis were known, that model assumed that the first North Americans were the people associated with sites containing Clovis points. With further research in Quarternary Studies, palynology and archaeology an academic consensus developed around the model of initial settlement. It held that the movement was by large game hunters (their vision of a Clovis Culture) employing a distinctive tool set and dated to a remarkably narrow timeframe around 13000 years ago.

With the discovery of new sites dating substantially earlier, notably Monte Verde and Meadowcroft Rockshelter, "Clovis First" and models of Clovis culture lost a lot of followers. The controversy is a product of disputes between those who still adhere to the Clovis First and those who argue for an earlier foray into the new world.

Edit: I made this comment solely to acknowledge that there was abundant info on this subject available in the FAQs when there was already a very popular top level answer posted. That answer has since been deleted along with some others. Now my comment is widowed here and may seem oddly out of place. Since then u/CommodoreCoCo has posted a superb response - please read it!!!

In addition to the great answer by /u/CommodoreCoco, I'd like to expand a little on the role modern molecular biology and genetics has played in making "Clovis-first" an essentially untenable position, in addition to just generally going into what it tells us about the ancient peopling of the Americas. The information in this comment is all taken from "Origin" by Jennifer Raff, a genetic anthropologist.

I'd like to lead by making a note about the ethical issues that come up with the collection of data from ancient archaeological sites which inevitably contain remains of people and cultures which are ancestral to indigenous peoples in the Americas. The history of curious archaeologists excavating sites without seeking consent or approval from modern native peoples who may object to the disturbance of their ancestors' remains is long and sordid, going all the way back to Thomas Jefferson. For a particularly egregious example, look into Aleš Hrdlička, a pre-eminent Physical Anthropologist and Curator of the Smithsonian Museum of Natural History in the early 20th century, who amassed a collection of tens of thousands of skeletons from around the world with little regard for ethical violations committed in the act of collection. Today, there are greater rules and regulations surrounding the excavation of ancestral populations, but many Indigenous groups remain reluctant to work with Anthropologists and Archaeologists for good reason, and unethical practices are still committed by less scrupulous researchers. Indigenous groups often have their own narratives to tell regarding their ancestors, who they were, and where they came from, and its important to respect these more traditional narratives. In fact, from the viewpoint of modern institutional science, researchers do well to take these narratives seriously, since they themselves can often contain useful information that can be incorporated to give us a fuller picture of what happened in the past.

So with that said, let's look into what contributions genetics has given to this picture. My answer will diverge a bit from /u/CommdoreCoco's, as I'll be focusing more on the bigger picture and less on specific sources of ancient DNA, but keep in mind that the "bigger picture" only comes from the examination and comparison of many genomes, each of which comes from a particular individual who lived in the past.

To start, there's the familiar picture. We've long suspected that most peoples in the America arrived via crossing the Bering land-bridge during the last glacial maximum. The genetic data strongly supports this, with very clear genetic links between Ancient DNA samples from northeast Asia and those from sites in Alaska. A fun detail is that we have additional data from the remains of ancient dogs in the Americas, again showing a strong signal of descent from Siberian dogs.

Genetic data from modern Native American populations also suggest that they have been isolated from other populations for a while, which would make sense. Although it is generally believed that Beringia was habitable by humans, there is currently debate over whether Beringia was merely a "land-bridge", just an avenue for reaching the Americas, or whether it was a site of long-term human habitation, with speculation that the southern coast of Beringia would provide a milder, tundra-like climate where humans could plausibly live. This is unlikely to be resolved until techniques are developed for the sampling of underwater sites.

Ancient DNA (aDNA) samples from sites across the Americas strongly suggest that the descendants of those who made the crossing later split into 3 different groups: Ancestral Native Americans (ANA), who make up the primary population who migrated southwards and are the ancestors of most Indigenous populations today; Ancient Beringians, who remained near Alaska/Beringia and are not related to any modern groups; and the aptly named Unsampled Population A, "known to us only indirectly from the traces of ancestry it contributed to some Mesoamerican populations."

Since they are the most relevant to the idea of Clovis/Pre-clovis, and are likely the main population involved in the peopling of the Americas I'll focus mostly from here on the ANA population. The ANA population itself split into two major genetically distinct groups after leaving Beringia, the Northern Native Americans (NNA) and the Southern Native Americans (SNA). As you may guess from the names, the former are believed to be ancestral to modern Native populations from Northern North America, and the latter to populations from South America and Southern North America. Importantly, this split seems to have occured about 15,000 years ago, after the ANA population left Beringia (The evidence for this is that both groups show similar levels of genetic relation to the Ancient Beringians, the group who didn't leave Alaska, suggesting that they migrated south together and split later. This is again supported by similar evidence from dog mitochondrial genomes.), but before geologists estimate when the "ice-free corridor" became a viable route for Southern migration, around 12,500 years ago, suggesting that some other route was used by the ancestral ANA population.

There is geologic evidence for a possible route down the west coast of Alaska and North America near the end of the Last Glacial Maximum which would have opened up a couple thousand years before the ice-free corridor. This would better match the genetic data, but unfortunately, again, any archeological sites would be underwater so it is difficult to test the theory of this coastal migration route. There is additional genetic evidence supporting a coastal migration route in the form of just how rapidly the SNA descendants populated southern North America and South America. aDNA Samples as far from each other as Brazil and Nevada, each from about 10,00 years ago, show a very close genetic relationship, and this is true of samples from many ancient sites. To quote Raff,

the pattern of population splits that the genomes reveal is so fast--nearly instantaneous--that the scientists who analyzed them likened the migration process as nearly jumping over large regions of the landscape. This fits more closely with southward migration by boat along the coast than with overland migration.

As of now, the exact route taken is still an open question, with, among other possibilities, some favoring the coastal migration route, others sticking to a revised hypothesis of the ice-free corridor, and some suggesting an even earlier Southern migration 25,000-30,000 years ago.

You'll notice I've mentioned Clovis/Pre-Clovis very little so far. This is because Clovis-first doesn't really denote a hypothesis of how or when the Americas were populated, but a hypothesis of who first populated the Americas, namely, groups of people belonging to the Clovis material culture. Of course, a how and a when were still needed, so the hypothesis of the "ice-free corridor" was developed as the main route for Clovis-culture individuals and groups southward, with the implication that migration began essentially as soon as a route was open. Really, as covered in other comments, strong archeological evidence of pre-Clovis sites belonging to other material cultures alone was enough to blast some rather large holes in the Clovis-first ship, and for a while "Clovis-first" itself has been a fairly fringe position. Very few experts today believe that the initial peopling of the Americas was carried out by those belonging to the Clovis material culture, and that's been true for a while.

However, there are still traces of the Clovis-first model to be found in those who continue to support the hypothesis of the ice-free-corridor as the primary southern migration route. I hope I've shown in this comment that the genetic evidence we have collected so far does not mesh very well with the hypothesis of the ice-free corridor, and instead suggests an alternate route, such as coastal migration. Of course, just like Clovis-first and the ice-free corridor, all of this is open to revision pending more data from archaeological sites, ancient genomes, and geological models, among others.

you may wanna check out r/askanthropology if you don't get a solid answer here.