I guessed that the button's shaft directly poked the switch
contacts. So I squirted some cleaner into the switches.

What brand of cleaner?

Clorox Toilet Bowl.

No, seriously... M G Chemical's Super Wash.

When you buy toilet bowl cleaner, look at the list of ingredients for
hydrochloric acid (or some strong acid). Cleaners without it don't do a very
good job. I kike Sno-bol.