Not so fast...
Well, it happened again. Car will start, but can't get it out of park and the brake lights are not working.
So, either the new switch went bad (what are the odds?) or there's some other (obviously intermittent) problem (maybe wiring?) that keeping the signal from getting where it needs to go.
I wiggled the connector around as best I could (I couldn't take it apart without tools), but no change.
Boy do I wish I'd held on to the old part so that I could test with that! (normally, I do that kind of thing, but this time I didn't because I thought the old switch was bad).