Small problem with this notion. Humans and our direct ancestors only came into existence after the last ice age. We did not exist prior to or during the peak of the last ice age.
All this says is that a group of Homo sapiens crossed into N America and then spread throughout N and S America. The idea Homo sapiens evolved in S America would mean that our species evolved in two different places ... which we did not.
I can tell you, as a student of both anthropology and archaeology, that you're talking in an awful lot of absolutes about a period of time (anything before say 5500 years ago) that we're lucky to find literal, specific, grains of sand in the deserts that remain of the interglacial bread baskets, to come up with those theories you're so confident about.
History, like science, evolves. Only ignorant people think it's written in stone, and that the translation is done by a precision instrument.
The fact that it wasn't written down at all, that's the only absolute.
Genetically, our species, homo sapiens, has existed for hundreds of thousands of years, and co existed with at least one other species of hominids BEFORE the beginning of the last ice age.
And that's just what the bones tell us.
The world's a big place, but nowhere near as vast as the eons that human beings have been walking around, motivated to move in order to survive.