there were horse...then the first Indians (we can call them Paleo-Americans) came...then there were no horses...then there were the Spanish..then there were horses...the Indians took to the idea of riding them at that time(and probably ate some too).
....whether or not the indians killed off the indigenous horses (ie. caused their extinction) is not provable....but finding horse bones with cut marks from stone tools gives science a pretty good idea that the earliest Paleo-Americans killed and ate horse (and mammoth, giant ground sloth, giant bison....etc)
The only reason why Indians took to riding horses was because they first saw whites riding them and thought--yeah, what a good idea. Ditto the lance. The Comanche were pretty good at each, and of course great a shooting arrows from a full gallop. It took W.T. Sherman himself finally to taken them down.