I don’t think it’s that deep. A child’s dad will say something about “chasing off the Indians” and their first instinct would be to make a game out of it because they like to run around. Children are like sponges, they’ll pick up anything in their environment without needing strict indoctrination.

pretty sure chasing games exist everywhere and are very old, unsure if "cowboys and indians" predates cowboy movies, or "cops and robbers" predates similar media