Ice was constantly getting caught in the disk-shaped ice door assembly. This would cause ice to pile up behind the door. Sometimes it would drop a huge amount of ice when the logjam cleared. Sometimes it would just get completed packed and have to be cleared manually.
The exact cause was a mystery at first. The solenoid was working and the ice door was opening, remaining open for a few seconds, and the closing when the ice dispenser button was pressed, as it should. However, ice was still getting caught when the door closed about 70% of the time.
Closer inspection revealed that the rubber that makes up the outer portion of the disk-shaped ice door was warped. This had initially unnoticed effects.
First, because the door was warped, the opening through which the ice fell wasn't wide enough, causing ice cubes to get caught and jamming the door in a partially open position.
Second, we were never getting a good seal. Warm air was always getting into the freezer through the ice dispenser door, resulting in excessive frost. This extra frost collected mostly around the ice maker, which compounded the first problem since ice built up inside the chute that the ice cubes fall through to be dispensed.
I knew that the warping of the rubber on the little door as a big factor. The dispenser door recess crank also looked warped and the plastic was worn down where it links to the ice dispenser solenoid. I also thought the door recess spring might have lost some of its springiness over time, resulting in less force pushing the ice door closed. Finally, I noticed that the linkage between the solenoid and the dispenser door recess crank was rusty and worn.
Replacing the door might have been enough. The refrigerator is very old though so I decided to just proactively replace the solenoid, spring, and dispenser door recess crank in addition to the door. Shipping isn't free and I didn't want to order parts one at a time. More importantly, my wife did not want to continue suffering with the problem while I replaced the parts one by one. This is a key consideration in any repair.
The parts are all easy to replace so I only have 4 tips:
1. Be sure to turn off the power or unplug the appliance. There's line voltage running to the control panel that you have to remove to get to these parts. You can get a shock if you try this repair without cutting power. You can also short out something on that control panel, resulting in needing to buy another part.
2. Be careful with the wires going to the control panel. They plug in with locking connectors and that plastic gets brittle. Be gentle unplugging them.
3. Note the path that the wires take through the other parts before unplugging them from the control board. You have to feed them through the same way or they'll get crushed or cut when you reassemble everything.
4. If you're new to DIY repairs and are afraid of taking the fridge apart and having trouble getting it back together then take cell phone pics at every step of disassembly. Then you don't have to worry about remembering where things go.