There are pretty strong theories that over hunting caused the extinction of many mega fauana at the end of the last ice age. Though obviously, the climate change caused disruptions that would allowed our ancestors to push many species over the edge.

And sure, modern domestic horses would be different than their wild ancestors, as they would have been less adapted to their new environment, and that can pose problems for the horses themselves. But 10,000 years is hardly time for ecosystems to adapted without them, in fact, eco systems are dying because we have killed off all the large herbivores. Have a video to explain things far better than i can