To deal with the stripped cleat screw problem mentioned earlier I headed over to Home Depot, encountered a toilet, then bought a Black & Decker Screw Extractor Set (#16270). Also known as “easy-outs”, these conical, left-hand threaded tap-like devices allow you to drill out the head (or center) of a screw and then thread the extractor down inside and remove the screw. These are useful when you strip the head of a screw, have rusty parts, break screw heads off, and other things like this. That is, they are useful when they don’t snap on first use.
I drilled out the hole, set the extractor in the hole, attached a tap handle, then started turning the extractor to embed it in the screw and break the screw loose. Right as I did this, there was a subtle POP and the whole tip of the extractor broke off and remained lodged in the hole.
While I was able to drill out the other screws to a slightly larger size and use the broken extractor, the first screw now had a bunch of very hard metal stuck in the center of it. This particularly hard steel thwarted all efforts at drilling with the bits I had on hand, so I had to slot the screw with a cut-off wheel, then go at it with a screwdriver. One very large screwdriver and a surprising amount of torque later and all four screws were out.