For the most part Indians did not domesticate the animals they ate. They hunted for their meat. The Europeans did domesticate their meat animals. In Europe they often lived in close proximity to them. Even keeping them below their living quarters in the winter. This close proximity meant that they adapted to animal diseases. Because immune to them. Those who did not left the gene pool.
Europeans brought these animals (pigs are a fine example) to North and South America. The natives immune systems were useless to counteract the diseases that came with these imported animals. The natives died, though some were strong enough to survive these imported diseases.
The Mexican and Central American Indians did not have large animals in any quantity to hunt. Unlike the US plains Indians with Buffalo, Elk, and Deer to hunt. Thus the idea of making “bread” of their neighbors.