If by small numbers you mean 100s of thousands to millions of the people than yeah they could have easily drove those animals extinct.
About 3 million years ago, Panama arose out the sea and connected previously separated North & South America. This allowed the Saber tooth tiger to enter South America where they immediately reeked havoc and drove pretty all of South America's Ungulates, marsupials and large predators like Terror Birds extinct.
Now if small numbers of saber tooth tigers acting individually can drive a large number of animals extinct then it's not much of a stretch to see groups of humans who compared to Saber tooths are much more efficient and diverse hunters doing the same.
Especially considering one of early man's hunting strategies was to cause stampedes and drive whole herds of animals right off the sides of cliffs.
Except there is zero evidence that the human population density in N. America was anything like that high 12,000 years ago.
"This allowed the Saber tooth tiger to enter South America where they immediately reeked havoc and drove pretty all of South America's Ungulates, marsupials and large predators like Terror Birds extinct."
Well, that's one theory, anyway; but it's in direct contrast to another theory that holds that smilodon died out because they were too specialized in their prey and became extinct when that prey died out. They may have thrown off those feeding inhibitions once they went south...
"considering one of early man's hunting strategies was to cause stampedes and drive whole herds of animals right off the sides of cliffs."
They were still using that trick with buffalo when the Europeans arrived, and yet, the buffalo herds stretched from horizon to horizon and reportedly would take days to pass by, despite having been "over hunted" for thousands of years.