Well, I squirted some oil into the spark plug hole, on the advice of The Internet. This did not fix the problem; in fact, it so thoroughly fouled the plug that I had to go out to get a new plug. Fortunately, that was on my troubleshooting checklist anyway. After installing the new plug, the mower fired up and... died after running for ten minutes.
If this mower wasn't on its last legs to begin with, this is where I'd take it to a small engine repair shop, because I'm out of troubleshooting ability. Instead, I think I'm going to take the $150-$200 that I would spend on a repair, and look for another mower-on-its-last-legs to buy from somebody else. My neighbor's got one parked in his barn for the last year that apparently has nothing wrong with it except it won't crank. Thankfully, starters are something I feel pretty confident troubleshooting (trace 12v from battery to starter, replace the thing that is not passing 12v), so maybe he'll sell it to me for a song.