This fix was easy so I'll talk about the diagnosis. When I tried to use the ice dispenser I could hear the motor running. I ruled out an issue with the screw mechanism in the ice bucket by turning it manually. Next, I tested the ice dispenser without the ice bucket in place by opening the freezer door, pressing the door switch and pressing the ice dispenser control on the outside of the door. To my surprise, I could see that the drive coupler was spinning. I didn't expect it to move and couldn't figure out why I wasn't getting ice.
I decided to get a look at it when all the parts were in place. I set my phone to take video, carefully placed it so it had a few of the dispenser drive coupler, put a flashlight next to it, closed the door, and tested again. I could hear the motor running but the auger didn't spin! Aha! I then took out the ice bucket and tried the first test again, only this time I gently held the dispenser drive coupler. The motor shaft turned but the drive coupler didn't. It was stripped. It had enough friction to turn when there was no load against it though.
The next steps were trivial. I found the parts using the excellent diagrams and images provided by this site, watched the instructional video, and ordered the parts. They arrived a day earlier than promised. The repair took me about one hour. My 9 year old daughter watched and helped.