Minor archaeology correction: The current scientific hypothesis shift is to multiple entries.
There were definitely people in the Americas before the Beringia land bridge opened up. I have been to and seen the human footprints alongside mammoths and giant ground sloths in the 21-thousand-year-old clay in New Mexico! There are well-established campsites in Chile that date to the same time or before the Alaskan/Canadian ice-free corridor was opening up! The ancestors of those people very likely came to the Americas by boat, following the Pacific shoreline.
However, it’s also true that a massive amount of a specific type of tool, called Clovis point, that starts showing up right when the Beringia land bridge ice-free corridor opens up ~12,000 years ago. Most archaeologists agree that there was a migration south from Beringia/Alaska at that point, a new group entering the Americas seen in increased populations and a new stone point style. The point isn’t that that didn’t happen, just that they weren’t the first time people ever set foot in America—there were already people when the Beringia-peoples came and integrated. (The Pawnee historian and writer Roger Echo-Hawk has an interesting perspective on that here.)
However it’s true that the causes of the Pleistocene megafauna extinctions aren’t fully understood, but the changing climate -> large grasslands becoming forests -> massive grazing herbivores didn’t have the food sources they used to is a big part of it. (Evidence: the places where there still ARE large grasslands, like the North American plains and the South American pampas, retained large herbivore megafauna)
And as my Ecology professor from the Confederated Salish and Kootenai Tribes once told us, when discussing this, and I never forgot: If it’s true that the people who came to the Americas in the Ice Age hunted the mammoths to extinction, maybe it was an important lesson our ancestors learned back then. Maybe that’s why we place such an important value on reciprocity, care, good stewardship, and relationship with the Earth and the animals: that we saw what happens when we don’t. There are different ways to think about this story.