The odds are phenomenal that we wouldn't.
Fossilization is rare.
Even remains are rare; in the US, human remains create a big fight (under NAGPRA) precisely because they are so rare. In the case of the tribes, there were different methods of dealing with the deceased, including excarnation (exposure to be eaten by animals).
Glaciation has happened for long periods, altering the landscape.
And the number one answer...
No one has even bothered to look, because, after all, everyone knows that a few horses lost by Coronado's expedition produced all the horses used by all the tribes which used horses.
Coronado wandered around circa 1540, and AFAIK no artifacts from his expedition have ever been found (and people have looked). Horses were widespread in use (particularly west of the Mississippi, but not exclusively so), so much so that the Lewis and Clark expedition took it for granted when they found members of tribes on horseback.
And the number one answer...
No one has even bothered to look, because, after all, everyone knows that a few horses lost by Coronado's expedition produced all the horses used by all the tribes which used horses.
Not entirely true. Before the Spanish reintroduction of horses to the Americas, there is no evidence of domesticated horses in the archaeological record. After the reintroduction, evidence of domestic horses among the native plains populations rapidly became endemic. Why the sudden change? The possibility that a hypothetical native population of horses could go from invisible obscurity to universality in the archaeological record so quickly and coincidentally to the reintroduction of horses by the Spanish stretches the limits of credibility.
The incredible usefulness of the domestic horse makes it inconceivable that native Americans would not have exploited such an resource, had they been available. Before the arrival of the Spanish and their horses, the people of the great plains managed to scrape a subsistence by chasing buffalo over cliffs. After the reintroduction of horses, the intelligent and resourceful people of the plains rapidly capitalized on the potential of this new resource. Instead of waiting and hoping for a buffalo herd to wander close enough to a usable "buffalo jump", the horse allowed the plains people to run down the buffalo at will. Almost overnight we see in the archaeological record a shift in native plains cultures and economies around the horse.
On the other hand, every tribe on the great plains at that time, I think, had recent memories of acquiring horses and moving onto the plains.