Let's try to be precise about what "horse culture" constitutes.
Pekka Hamalainan's Comanche Empire describes the kind of horse revolution I believe you're talking about, where horses were a measure of wealth Native people herded, traded, and raided. The horse for them was their means of developing something a lot like an imperial power where their sovereignty stretched from Oklahoma down to Northern Mexico. Theirs was a hemispheric power, and other Native groups had no such hegemony.
Daniel Richter has called on us to "face East from Indian country" and to ask these exact kinds of questions about differences among indigenous groups. There was no "they" historically, at least not until the ghost dance movement unified so many Plains Indians into an alliance of convenience. But Richter also warns against bringing the same Western categories of analysis. Horse culture is not something that merely turned on or off.
Elizabeth Fenn recounts the Mandan relationship with horses in Encounters at the Heart of the World. She found plenty of evidence Mandans had horses for meat and trade, but they never took to the equestrian practice of riding them. Why? Not out of any failure of imagination! The Mandan villages were one of the few sedentary, non-nomadic groups of the Plains. Their culture had formed well before horse culture drew Native groups out onto the Plains after the Colombian Exchange. The Mandan riparian life along rivers did not need to change, at least not until the pox arrived and killed most of them. Chalking that outcome up to horselessness would be a mistake, imho.
Could you clarify what you mean when you write "Daniel Richter has called on us to 'face East from Indian country' and to ask these exact kinds of questions about differences among indigenous groups." From context I think you're saying Richter is suggesting that examining the development of horse culture in a lens specific to native American groups isn't really the right way to look at these developments? That instead it's more appropriate to view those changes in a way that a more "generic" group might incorporate orutilize the changes involved with acquiring horses?
let me know if I'm completely off base or not. Thank you for your reply and to OP for the question. It's a paradigm shift I really hadn't thought much about before.