Success at last! I got the VC Trigger Generator working, finally.
The culprit ended up being the UJT. However, given that this was a $10 part, I wanted to make certain that it was at fault before replacing it! David Ingebretsen helped me with understanding and debugging this circuit - thanks so much! I learned a lot in doing it too. Lots of measuring voltages at various points, particularly transistor bases. Eventually figured out which blocks were working, and which weren't. Narrowed it down to the UJT, popped it out, put a new one in, and presto!!
Calibration was a bit of a pain, due to the fact that I'd moved the trimpots all around, but it was all set after about 5-10 minutes in front of the oscilloscope. The Duration/Width pot was working the opposite way of how I'd expected it to work, with 10 being the smallest pulse, and 0 being the widest. Apparently this is how David's works as well, it is how it is noted in the schematic, and Nyle didn't notice anything odd with David's unit... so presumably it is how it is supposed to be? Anyhow, it seems a bit odd to me, so I switched the MTA and reversed it.
Another oddity is the fact that at rest it outputs approx 10V, rather than floating - this is due to the way that the LED is set up. On its own this isn't a problem, but I am a bit concerned about OR-ing trigger signals from these modules. It doesn't look like it should be an issue with the other trigger generators (i.e. the Sequencer and Peak Selector).
Also, note that I didn't screw the PCB down all the way onto the top two standoffs - the pushbutton switches are rather large, and push against the board. If I were to do it over, I'd just put the wires going to the pushbuttons going off to the side, rather than straight out.
It really has a lot of range - I didn't measure it, but with the range knob at 12:00 the pulses have a period of 85ms, and from that it goes really fast, and really slow. I didn't try the 3 inputs yet, but given the circuit and that everything else works, those should work without issue.
A great way to start the new year by getting this working!