This was a 12 year old problem.
When the light first burned out in 2003, I tried changing the bulb with no luck. Then, I removed the socket and tested the current--0. Unfortunately, that's where I stopped the testing. We moved the refrigerator downstairs and bought a new one.
Then, the same thing happened last month to the new refrigerator. I found the your video on replacing the door switch (duh). And fixed problem number 1.
Then, I moved down to the basement and changed the switch and replaced the socket and bulb. Voila, let there be light. Too bad I didn't try that back in 2003...