Yes, when one is used to running every where you go, one has little use for a horse. Except for specific duties like hauling a travoise, or getting you up close to a running bison to deliver a lance. But even then, it was easier to just drive the bison over the edge of a cliff.
A good athlete, and the American Indian was every bit of that, just does not need much of a head start on a horseman. And he will never be caught.
Note how closely relay stations were spaced for the Pony Express.
