I am aware that this is just the impression that I have from watching movies, games and series like Westworld. So I wouldn't be surprised if this is just a movie trope and has no basis in actual history.
But after the Europeans arrived in America and the horse was (re)introduced to North-America it seems that at least some groups/tribes adopted the horse. But somehow they didn't adopt the saddle and stirrup combo that the Europeans used on their horses. Is there a particular reason why they didn't use saddles?
There is some truth to it, though not a sweeping generalization of all tribes.
What you are referring to is riding with a "war bridle." War bridles were indeed popular due to being much cheaper and requiring less resources than a full saddle apparatus, and could be fashioned from discarded reigns. (A war bridle is basically a leather strap put in the lower jaw of a horse held by the rider with no headstall to attach reigns to.)
As to why? This probably would depend on the tribes resources or trading standing with various groups. Eventually the saddle/stirrup/reign setup was the norm.
A book called "The Indian war of 1864" By Eugene F. Ware published in 1911 might be some reading that may interest you on the overall topic.
Hope this helps a little.