Don't think that's it. If I press the brake, it locks the disk immediately. No slow response or weak breaking power. No pumping necessary. However, there's still some play back and forth.

What I mean is that if the brakes are applied and released repeatedly, the wiggle only occurs on the first locking in either direction. Change direction, and it wiggles the first time the brake locks - and only the first. Keep the brake pressed and move the wheel back and forth, and it wiggles every time.

It seems whatever-is-loose stays in its forward or backward position until it's pushed to the other.

Brake pads are still suspected. We'll look into it next week.