Tens of thousands of people have literally walked across the US continent. Ancient people walking across Eurasia doesn't seem that worthy of surprise, on the contrary, it would seem surprising if they didn't.
Let's ask a question. Is it known whether anyone walked across North or South American in the 1600s? Native or European?
They covered a tremendous amount of ground and, out of about 600 who started, only 3 or so survived. The most famous is Cabeza de Vaca, who would return to Spain, write a book, then be sent to what is now Paraguay and Argentina, where he governed for a while before being sent back to Spain again.
He did all that without cars or planes, or motorboats. These people could go very far.
My favorite from the expedition was a Morrocan named “Estevanico” [2]. His story is pretty amazing.
My guess is that lots of humans traveled like this, over great distances. Some went far, some not so much. The North American tribes took slaves, traded them amongst each other, or sometimes seemed to accept strangers with open arms before nicely sending them along. Other times they would just kill strangers, sometimes almost for sport or entertainment.
Traveling back then wasn’t easy, and encountering other humans could just as easily accelerate your progress as it would stop you dead in your tracks.
But for nomadic hunters, there seems no reason why it wouldn't be possible in theory provided they didn't have to cross a desert or impassable mountain range. Which are serious problems for walking across all the continents.
Yet people did it anyway.