I believe that's 3J7 for the general illumination relay AKA solenoid #11.

Without diving deep into it I can't tell you EXACTLY what would happen with it backwards but I actually don't think anything super major should occur since you'd simply just be switching the +34V and the solenoid ground across the GI relay coil which happens to be found ON the power supply board. Essentially this is a controlled 'solenoid #11 input' to the power driver board, not an input/output of raw power as one might fear.

The bonus here is the GI relay itself wouldn't really care since it's just driven by another DC coil. Worst case is you might damage the flyback diode or blow the solenoid fuse on the Power Supply due to a short thought the flyback diode, or worst case blow the GI relay transistor on the MPU since the blocking diode would be backwards and might be a better path to ground than through the relay coil. This would kinda depend on whether the GI relay is normally closed vs normally open operation. This I don't quite know but I presume it is normally closed as the GI circuit is on when the power is on regardless of any input front he MPU as far as I know. This would make the above flyback diode discussion only relevant when the GI is switched on and off by the MPU and would thus be irrelevant in your case since you can't even start a game.

I personally doubt this plug reversal has caused anything as major as your issues. It's likely in your fixes or somewhere else entirely imo.

3J7_(resized).JPG
Relay_(resized).JPG

It is. He's talking about the power supply. Not the MPU or coil drivers.

No there's nothing here regarding troubleshooting Sys 7. That's just the pic he used to show what board and plug he was talking about:

All that era of Williams machines used similar looking power supplies, but it's Sys 11 through and through in this case (at least as far as this single connector goes).