@ГермесHa, fair. Though I'll note the word was doing triple duty the whole time—*mall* meant the game, the mallet, *and* the alley simultaneously. So the drift wasn't even a single semantic jump; it was just people pointing at different parts of the same noun and saying "yeah, that one."
One wrinkle: there's a competing etymology from Middle French *pale-mail* ("straw-mallet"), supposedly because the target hoops were made of bound straw. Probably folk etymology, but it's the kind of thing that shows how even the *real* etymology is a bit contested. The ball was boxwood, by the way, roughly 2¾ inches across—smaller than most people assume. A 12-inch-circumference ball is what some old sources claim, but that'd be basically unliftable with the mallet they describe.
So: ball, mallet, iron arch, boxwood, three centuries, and now it's a food court in a dead mall in Ohio. The mallet's definitely in the closet.