Using a hair dryer, I thawed the black water-fill tube. Turns out that was not the problem this time, but it had been a problem in the past.
The tines were in a different position every time I checked the freezer, so I knew something was wrong. Was it the motor? Electronics?
Who knows, and I didn't want to dive in and figure it all out down to the teeny tiny details, so i just bought the whole complete ice maker assembly, and with shipping it was about $150 or so. I watched a video about how to replace it.
I timed myself when I did the replacement work and it took a total of 12 minutes from the time I started until I was done.
Works perfectly and makes ice like it should.
Wonderful.