Well, I'm just now realizing I didn't take any photos, but last week I spent a few hours with my dad fixing this.
Can't confirm the car runs (it's been sitting and he didn't have jumper cables and I only had a rental car), but I can confirm the shifter works. Actually probably works better than it ever has during his ownership!
This is an image my dad took while I was trying to diagnose over the phone a few months ago. The issue can't quite be seen well here but it will give you an idea. Now that I know what the issue was I can see it in this photo but I couldn't see it with only the photo.
If you look closely you can see a rope just behind and to the right of the spark plug boot. Just behind that rope is a bell crank that is part of the shifter assembly (it's the 1-2 bellcrank, if you're curious). Literally right above the rope is a ball attached to the bell crank. And the rod end that is supposed to go on the ball is not really connected.
There is supposed to be a rubber bushing in there and a clip to hold it together. Since the end can fall off the ball (and has tons of slop even if it doesn't fall off) the shifter can easily get stuck in 1st or 2nd and then the lever just flops around without moving the transmission.
Solution?
Go across the street to the local flaps and buy an assortment of rubber pieces. Vacuum nipples, pcv fittings, even valve stems. Find one that looks close, cut to adjust with knife, press back in.
The second pvc fitting I tried worked nearly perfectly! Tight fit on both the ball and the rod end, and holds the stuff in alignment.
But all this was done in Denver on the day before we got 12+ inches of snow, in dad's apartment parking lot, with my dad's old tool set (which to give you an idea includes only Robertson screwdrivers because he is Canadian and has lost all the normal ones).
Now I know the issue, so I think I can order the right parts to fix it the right way. But if he had to move the car now, he'd be able to freshen the gas, jump the battery, and at least drive it somewhere.
Progress!!