You could argue that American football plays quite a bit like rugby football, Gaelic football, and Aussie Rules football though. Association football is the odd one that decided you literally can’t use your hands oh except the keeper and oh I guess throw-ins? Every other form of football involves handling the ball with your hands, including the older forms from which modern ones descended.

I think you’re looking at it through a modernist lens; a lens through which the role of horses is virtually nonexistent, and you have exposure to a wide range of international sports with different lineages. Basketball and handball are much newer than the concept of “football,” and share no history with it, so it’s no surprise that they didn’t wind up being called “football.”

The claim isn’t that everything played on foot should be called football (that would be a weird criterion, and not useful). The claim is that the group of sports called football are so called because they are played on foot, not because players are only allowed to use their feet.

It’s not a super widespread idea, but Wikipedia discusses it, so it’s at least not just something I made up.