I took another crack at it tonight. I can't see anything tweaked on it. It's as straight as a slightly floppy piece of nylon can be expected to be.

It can't be in another position as best I can tell. The nylon piece is screwed onto a flat "bar" that goes all the way to the top of the machine and there's no room for adjustment at the screw or any obvious places further up the chain. The other end is held in place by a terrible setup using pieces of flat metal bent into weird shapes and screwed to the frame.
I set the timing, and I reassembled, again, tested by hand - it caught the thread, but didn't let it go again, and as soon as I ran it, it jumped out of time. It won't pick up the bobbin thread anymore, and I can feel that the gear is sitting in the wrong position (handle hard to turn in one spot, but still turns so not as far off as when she brought it to me)
I think it's the weird jerry-rigged side of the nylon gear that's letting it jump time. Typically it jumps one tooth.
I have a sneaking suspicion either there's a piece missing from that nylon part on the jerry-rigged side (nothing dropped out, and it -looks- smooth, but I suppose it's possible for a piece to have broken off and gone through my sewing room's black hole into someone else's sewing room?) or something elsewhere is bent or otherwise ruined.
After my 6th time reassembling and timing, etc, I'm wondering if it -IS- possible to do so much damage that a machine can't hold time anymore. I never bought into that "timing mechanism is worn out", "won't hold a time", etc, but I'm starting to wonder. This isn't my first visit to the Timing rodeo... I would just think it would show
something if there was so much damage.
Additionally, without touching anything, the take up lever seems less "off" today. I think this machine just wants to be put to rest. I'd sure rather that wasn't the case though. I'm still open to ideas, I can strip this machine down to it's bare frame within about 5 minutes now. *sigh*