That's true, but I suspect all humans needed to wipe most of these animals out was the invention of the spear. There's a difference in what humans hunt compared to most animals. Large healthy animals are quite dangerous, as anyone who's been kicked by a horse can attest, so often their predators are limited to the sick or otherwise vulnerable ones, limiting the population damage they can cause.

Humans can throw a bunch of spears at the biggest thing they can find and follow it until its injuries to do it in. That's great from the perspective of being able to get a lot of food, but not so great in that you're killing off the healthy breeding population.