Since I got some good suggestions here, I thought I'd post the ultimate resolution of the saga:
I finally broke down & took it to a local shop. They let it run for an hour, but it never shut off. So the next day, they went for an hour and a half. Still fine. I suggested that it probably needed to be under a bigger load, and they should
cut some grass until it quit.
That was a good idea, except that in running it so long on previous days, a hole in the muffler melted the deck belt. Okay, a few more days to get that fixed, then they were finally able to reproduce the problem.
When it quit, his first thought was ignition coil. (Everyone who suggested that was right.) But it was a bit worse than that; I had a mouse nest in behind the blower cover, and they'd chewed the insulation off of everything.
The shop pulled the engine & fixed it all. $427 (including the new belt & muffler repair.). Ouch; but I was able to
cut the entire lawn at one time for the first time all year.