How differently do you think American Indian civilizations would have developed if they'd had horses from the beginning?

Actually, horses evolved in the Americas, but died out about 7,500 years ago.
The Indians, as we know them, didn’t have them until the 1500’s, when Spanish galleons wrecked on American shores. Some horses survived and made their way to shore. From there the Indians quickly domesticated them, and they quickly became a major part of their culture, and I’m sure they made huge strides in their civilizations during the 300 years they had them, before the Europeans came.

What, in your opinion, would have been different if they’d had them for the last 20,000 years? Or last 10,000 years?

Is it possible that the domestication of the horses is what led to world dominance?