I found a second 12voltvids video with a JVC unit with the same mechanism that won't rewind. It's also the second JVC related video where he does not revisit the issue with followup video with a fix.
https://www.youtube.com/watch?v=__PFQlKia68
However, there're quite a few helpful tips in the comments section. One user suggested it could be a warped "pendulum." I don't know where that part would be. Another user said their photoconductor had bad contacts and was causing noise on the signal, making it not rewind correctly, and offered a suggestion for a fix.
So I have some new avenues to explore for troubleshooting. Could still be electrical, but could also be a mechanical issue. I'm going to try to replicate what he was doing to get it to rewind reliably without a tape.
The noisy photoconductor has me intrigued, though, because I initially thought it was the sensor and seemed to get it to reliably rewind when I covered it. But then later it wouldn't rewind even if I did cover it. So if it's noisy, it could be that the first time I covered it, it was
just enough to get it to work. Then it got more noisy the next day, and covering it wouldn't make any difference.
Anyway, just an idea.