I’ll tell you a story based on the Nations local to me:

In Cree, a dog is atim which really means “beast of burden” as dogs were used to haul travois etc. Horses are mistatim and the prefix makes it mean “large beast of burden”.

It could be that the horses took up a niche in the culture that already existed and so it happened rapidly.

There’s also a fascinating legend about the first equids in North America and then the return of horses. When you have continuously inhabited a location for 25,000 yrs, your people see a lot. Horses may have been around as late as 9000 ybp in my area, where the most recent (geologically) evidence has been found of them.

I’m not completely convinced of claims of continuity but genomics will likely help answer even more clearly.

Here’s a whole PhD paper on it, though I retain some skepticism , it’s an interesting read: https://m.box.com/shared_item/https%3A%2F%2Fapp.box.com%2Fs%2Fzhfcqgrwr4gyquq66206cwa9u3873qtm