According to this article, the shipwreck of a Spanish Galleon off the coast of Virginia that resulted in the Chincoteague ponies on Assateague Island was much later than the era we are talking about - 1750
https://www.nationalgeographic.com/animals/article/famous-chincoteague-ponies-may-actually-descend-from-a-spanish-shipwreck
A year or two ago I went down this same huge rabbit hole about whether or not there were living pre-Spanish horses in North America. The fossil evidence of the early horse ancestor Eohippus is pretty well documented to have evolved in the Americas and then spread across the land bridge to Asia and Europe, the opposite of human evolution and migration - and the horse spread into Asia and Europe much earlier than humans spread into the Americas.
I read the entire PhD thesis that argues that the Plains Horse Tribes did not learn horsemanship from the Spanish, and it is actually quite convincing. It was written by a Native American and interviewed multiple other Native Americans about their oral traditions, and goes through an extensive lit review (although more from a sociological slant than a science slant). It argues that history (and science) is dominated by European white culture (true) and that the Spanish insisted that the existing peoples of the Americas were less superior than Europeans, hence why it was OK to colonize and build missions to convert them to Catholicism and exploit their resources and labor.
The primary source of the PhD thesis is worth the read.
The thing that got me the most was the timeline - Columbus landed in the Carribbean in 1492, and the Spanish Conquistador Hernan Cortes landed in mainland North America (Mexico) in 1519 with 500 men and 15 horses, marched to Tenochititlan (Mexico City). Then⦠in 1522 (so three years later) a Spanish slave trader and captain reported seeing native peoples with horses off the coast of Georgia and the Carolinas. There are no records of those 15 horses being stolen (we have journals from one of Cortesās soliders, written in Spanish, about his conquest of Mexico. And if one of their warhorses had been stolen it would have been recorded. Those horses were weapons of war and pivotal to how they conquered the Aztec nation and Moctezuma). There are many legends in Native American culture about āElk-Dogsā - which this PhD thesis argues were horses.
It is worth a read, especially when you consider the amount of censoring and suppression of Indigenous culture through history.
https://www.proquest.com/openview/1b40a9128eaba2e22ab3fed4cf6551a8/1?pq-origsite=gscholar&cbl=18750