The overlap between the last known modern horse and the first people in America is about 2000 years. If they domesticated horses in that time, how come the hors

The overlap between the last known modern horse and the first people in America is about 2000 years. If they domesticated horses in that time, how come the horses didn't thrive along with the human population? I think the horse population where already on the brink of vanishing due to climate change after the last ice age, and the new Americans simply ate the last of them.