Well, it turned out that the insides of the faucet mechanism were shot. There’s a notch that’s supposed to keep the ball in place, and it had worn away.
Still, nothing that $100 of new faucet can’t fix.
Oh, and as with the toilet, the pipes needed to be hacksawed off, along with the bolts, as the entire thing had rusted up years ago. Another job I could never have succeeded at. The plumber said it was the worst rusted-in faucet he’d ever seen.