Football distinguishing it from games played on horseback

The problem with this line of argument is that basketball would also be football, as would baseball, cricket, hockey, and even cycling.

I'm not beholden to what some rich kids (not my class) of the university of Rugby or Oxford or whatever (not my town) in England (not my country) called the sport once upon a time.

Same. I don't remember when I had the urge to check soccer/football term origins and was pleasantly surprised about it. IIRC it also have some class segregation about it as well (of course it does).

longer than football (soccer) was called just football in Britain

This may blow your mind but there are other countries and they have always called it football (or the localized equivalent).

So basketball, baseball and handball is basically football too, neat.

Are you proposing to renaming polo to horse golf? This logic is a bit weird but sure.

That's not necessarily why, the origins of the word are lost to the mists of time. Either might be true!

Why can't it be HandOlive