I followed directions I found on the internet (ruled out loose or corroded connections, verified no frayed wiring) and determined that it was almost certainly a rotor position sensor (aka hall sensor) that needed to be replaced.
Ordered the part (which arrived promptly), and followed the video instructions. Replaced the part, 13 year old machine is back to like-new working order.
There were three things that slowed my project: 1. I ruled out other possible problems rather than jumping immediately to the most likely solution, 2. the stator wiring harness was unusually difficult to disconnect (I have a very strong grip, but I ended up having to use vice grips), and 3. I did not have the replacement part in hand when I started my project. If none of those things had been an issue, I would have been able to do the repair in probably about half an hour.