OK, so
@irepairsega's comment about losing sync got me thinking and I decided to dig in a little more. Research here on KLOV led me to looking at IC91 (74LS125 buffer IC) and IC106 (the big custom) on the video board. I saw several other posts about similar problems that pointed to IC91 failing.
I was able to use my good Outrun Turbo conversion board as a baseline test since that was working. I attached pin grabbers to pins 5 & 6 of IC91 which are the A2/Y2 pair according to the datasheet. Pin 5 is the incoming composite sync signal from IC106, and Pin 6 is the output from IC91 to the video connector. I then put the CPU board back on the stack and powered the game up. Video was fine (this is the board set that works, so no surprise).
First thing I did was try my logic probe on the video connector's pins, with the following results:
Pin 5 (video ground) ==> LOW - duh
Pins 1, 2, 3 (R, G, and B) ==> "data" patterns - high speed irregular tone variations.
Pin 4 (Sync) ==> A very regular pattern. A high pitched squeal with a periodic pause. I would be pretty sure I was hearing a representation of the horizontal sweep and the periodic vertical blanking.
I moved the logic probe to IC91 pin 6 (sync output) and the pattern sounded the same, and also on IC91 pin 5 (sync input from IC106) and it also sounded the same. I then decided to try the $30 pocket oscilloscope I grabbed from Amazon (who knew you could get a basic o-scope for $30? I had no idea!). The video connector (and IC91 pin 6) both showed this:

3.3v RMS and 2.9Vpp (seems consistent with my understanding of typical monitor voltage levels) and a 15.438KHz frequency. You can see the periodic dropout for the vertical blanking.
I moved my probe to IC91 pin 5, and saw this:
Similar frequency and cycle time, higher voltage. I guess the IC106 chip operates at a higher voltage level and IC91 knocks that down to monitor signal voltage levels, and inverts it as well I believe.
OK, so now I moved my probes to the same locations on the problem Outrun video board. My logic probe gave me different results:
Pin 5 (video gnd) ==> LOW (again, duh)
Pins 1-3 (RGB) ==> Data sounds (OK)
Pin 4 (Sync) ==> STUCK HIGH!!!
Sync was not pulsing, it was just stuck on a constant high signal. I moved the probe to IC91 Pin 6 (sync output) and it also read HIGH with no pulsing. Then I moved it to IC91 Pin 5 (the sync input coming from IC106) — and it was the regular pulsing pattern just like I heard on the good board! A ha! This points to IC91 being bad.
I also viewed the signal on the oscilloscope again. Here's the Sync signal from the video connector and also from IC91 Pin 6:

No pattern indicating periodic vertical blanking, and the frequency was reading 270KHz. Voltage was a fixed 3.6v, nothing to indicate pulses or a pattern of any kind.
Moved the o-scope probe to IC91 Pin 5 (the incoming sync from IC106), and it looked identical to the pattern from the good Outrun Turbo board:

Very, very similar values to the known good Outrun Turbo board.
Conclusions:
IC106 (the big custom chip) is good, as it's outputting a good sync signal just like the one on the known good board
IC91 (74LS125) seems to have failed, it has correct signal input on A2/Pin 5, but the output on Y2/Pin 6 is stuck HIGH and is not providing a 15KHz signal.
Plan:
Remove IC91, install a socket and a new equivalent replacement.
I have already ordered some 14 pin sockets and replacement 74LS125s (TI SN74LS125ANs specifically) from Mouser, and they will be here in a few days.
Will update in a few days. If I replace IC91 and still have a problem, then I will take it to see
@parism, but I'm feeling pretty confident I found the culprit. And I learned a bunch about tracing the video signal and using an oscilloscope too!