I have indicated here in green the responsible party. That ball valve was fused to the seat with rust. In fact, we'd taken this very housing off before to "fix" the problem, and never even realized all these pieces were in there! They had rusted into position. I cleaned them off, and replaced the ball (I was worried the line of rust around it might stick again) and put it all back together again. At first it worked but leaked, and then I replaced one of the plastic rings, and that stopped the leak...