The “American” horse wasn’t something to be ridden, the entire Plains Indian culture revolving around the horse started after the introduction of the Spanish bloodlines.
I know it’s popular now to talk about millions and millions of Indians living peacefully in harmony with nature and the rest of it until the evil Whitey showed up, but it just isn’t even remotely plausible much less possible.
That was 12 thousand years prior when the American horse roam the land.
There never was a peaceful time in the Americas with constant warfare, taking of slaves etc, plus massive floods along the Mississippi and elsewhere. The Mound Builders were the most prolific in east and central US, but they were likely wiped out by the invading Lenape.