Earlier quoted context omitted.
> It seems unlikely that King Edward III meant to ban games that involved walking on your hands, or perhaps brachiating. But this argument assumes that the etymology of 'handball' must be the same as 'football'. Football could be called that because you play it on foot, and handball could simultaneously be called that because you use your hand to strike the ball. There's no contradiction there.
No, all the argument assumes is that handball (and hockey) are played on foot. That would be enough for them to be called "football" under the theory that football is a class of games that aren't played on horseback.
If you followed this argument you'd be confused why we have basketball when netball already covers games that involve a net and a ball. Not all games are named by the same person with the same naming ideas.