This is an interesting question, not only because of the topic, but because of the certainty with which the statement in the thread title is made.

To be clear, the archaeology is not in any way clear enough about the early population history of the Americas to the extent that the following parts (bolded) in the thread title could be considered "settled."

The first humans to arrive in the Americas contended with giant sloths, mastodons, mammoths, and lots of ice. They thrived for centuries in harsh conditions dying out and being replaced by other migrants. Why did their numbers remain small, and what killed them off?

The question of when people first reached the Americas remains one of the more dominant research interests among North American archaeologists. The last three decades have seen a fair amount of shift on the "consensus" view (such as it is) of this hemisphere's population history. We've seen most professional / practicing archaeologists move from argumentative disagreement to grudging acceptance to welcoming embrace of new information that increases the early time-frame for entry into the Americas from Eastern Asia.

With each new well-dated site (that's the key-- without solid dates, you don't really have a good contender for an early site), the earliest date seems to be pushed back. The White Sands footprints are some of the earliest widely accepted at about 22,000 to 23,000 years BP. It's worth noting that the publication of that info is fairly recent, though, and it's very likely that the small contingent of remaining true skeptics of anything prior to Clovis (mainly Stu Fiedel and Julie Morrow at this point, I think) are working on an acerbically-worded response even as I type this.

But even without White Sands, the more recent older dates aren't really controversial anymore. At Coopers Ferry (Idaho) and at Page-Ladson (Florida) especially, dates of ca. 16,000 and 14,500 BP (respectively) are pretty incontrovertible, coming as they do from well dated contexts with obvious human-made artifacts.

So, let's assume that we have humans in the Americas at the earliest sometime between 25,000 and 16,000 years ago, probably on the higher end of that. I think we can safely agree to that with the existing data.

What we don't have is the following:

Any information about the biology of that population from which to conclude that it is not contiguous with the earliest biological / genetically sequenced individual, the Anzick-1 Clovis infant.

I'll repeat that. There is no evidence that the earliest American immigrants "died out" and were replaced. In fact, we have no idea if the "earliest" Americans are or are not contiguous with later populations, including Clovis. Some may not be, some may have been. As I said, the population history of the Americas is still something of an active research topic.

What we do have is genetic evidence of multiple waves of migration at different times in history. But according to the sequencing, Anzick-1 (which was Clovis, as well) does have later relatives, so at the very least we know Clovis lived on in its descendants.

So for the most part, this kind of nullifies the rest of the question in the thread title. As far as we know, nothing "killed off" the earliest immigrants to the Americas.

Also, I've read that these first arrivals left remains that appear somewhat different than later arrivals.

Earlier populations in the Americas-- as in any other part of the world-- don't necessarily look like modern populations, or like later populations. Human phenotypes are fairly plastic, and our species looks today differently from how it looked only a few thousand years ago.

The battle over Kennewick started because someone (Jim Chatters) made the ridiculous claim that the Kennewick skull looked "Caucasoid." Genetic testing some years later clearly indicated that he was, in fact, 100% related to the indigenous Americans who live in the region where he was found.

Were they from Europe rather than Asia?

It's an interesting, and somewhat bizarre, leap to go from "they looked somewhat different" to "were they from Europe?"

There is no evidence-- not one scrap-- that any immigrant population to the Americas before the 16th century had European ancestry. You could perhaps point to the Northern Europeans (Vikings) who tried to set up a settlement in L'Anse Aux Meadows, but they failed to gain a foothold and so can't be considered "immigrants" in my view.