Well, I managed to surprise even myself, and was able to re-assemble the shutter, the aperture, and put them back together, and everything appears to be working (including the self-timer)! I'm now working on reassembling the various lens parts, which turns out to be just as much of a challenge. I also realized that I have this piece on my work bench, and I have no idea where it could be coming from, which is never a good sign
I've looked at all the reference photo/videos I took, and can't spot it anywhere.