We've done alot of digging, alot of construction near rivers, alot of agriculture on the great plains.
Bones?
Teeth?
If there were horses here, the odds are phenomenal we'd have fossil evidence.
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.