So with the test switch held in the displays are working correctly?

Is this an original MPU board or a newer aftermarket replacement?

If it's an original MPU, you might want to try swapping U10/U11 ICs around. It's odd if you're saying that activating the self test switch allows the displays to function correctly. First thought I had is if the momentary switch is a NC (normally closed) instead of NO (normally open) variety. But it could be something faulty with the U10 PIA chip too. Half the display signals are going through U10, so it would explain why you're not getting anything on the displays (CA2 control line at pin 39 on U10 affects display blanking, display latch and display segment data).

Something else you could try just to see if it changes anything is unplugging J3 (cabinet switches) which include the self test switch, once you've started up a game and without the self-test switch activated. That would remove the self test switch from the equation temporarily just to see if the display behavior changes.

The PIA (peripheral interface adapter) chips aren't game-specific. You can swap those out of any board. They basically expand the number of inputs/outputs the CPU can control.

If you swap them back does it boot up as it did before?

If so, I'd say you've identified the problem chip. Hopefully a replacement 6820/6821 PIA would then get you working again.