Success. (so far.....)
I ended up doing the mod I described above, where I replaced the original LM2940CT generating the +5VD rail with a MIC29302WT adjustable regulator.
I set the MIC29302WT output to 5.10V, and the whole supply works fine now. (The original LM2940CT had an output spec of +4.85 to +5.15, so I don't see this causing any issues with the other parts of the system).
External VFD display turns on with good brightness (The 5.1V pushes the display MC34064 voltage monitor into the region that's full on, so that problem goes away as well).
Digital board powers up every time. I tested the output of channel1 and it's still accurate. It's been put back together running in it's home on the bench all morning with no issues.
In my desire to be done with this project, I didn't run some additional experiments that I probably should have:
1) I didn't use the pot to drop the output voltage down to the 4.95V that the original LM2940CT supply was making when the unit wouldn't turn on consistently.
2) I didn't just replace the LM2940CT with another LM2940CT and see if that fixed it as well.
3) I didn't replace the MC34064 on the display board (even though I'm suspecting that the 5V reg was causing this strange issues as well somehow).
If I had to guess, I think the problem was most likely just a faulty LM2940CT (even though the output looked fine on the scope). If anyone else has a 2302/2306/2308 that isn't powering up, I would start by just replacing the LM2940CT.