Thanks for the suggestions. I cleaned every electrical contact, including the grounds, then shorted the starter cable straight to the battery, and the engine cranked great, for the few seconds I gave the starter power. I thought the problem had been solved. When I tried to crank it again the starter would engage but the engine would not turn. I put a wrench on the crank and turned it myself, and every other rotation was much more difficult than the others. I turned the crank past that point again and fired the starter, and the engine turned twice and stopped at that difficult spot again. Too much compression once the rings got a little wet with oil and/or fuel? (If I ever get this thing cranking again I'll measure compression.) Not enough starter? The battery is rated for 330 CCA, which is considerably more than what I have on my Craftsman riding mower/tractor.
I tried the run-up idea, turning the crank until just past the difficult spot, and that didn't help.