that is an IDC (insulation displacement connector) and IMO they are crap

if you're sure you want to change it, i suggest using a crimp pin type

you may just need to slide the black cover off and if you have the insertion tool great, but if not try pressing on the wires with a small screw driver, once you take the cover off you'll understand.

assuming your running gas displays, have you checked the test points on the player 1 display, ground, +5VDC and about 185VDC depending what you set the output at on the solenoid driver board?

Your voltages seem ok, 173VDC is on the low side but it is preferred by some as it keeps the displays alive longer but puts stress on the 2 transistors in HV section. I learnt that from Quench

The commands and digit enables etc. come from J1 on the mpu board. You could try sliding the cover off and pushing on the wires in the pins?
Also checking the rear of the mpu board where the headers are soldered, looking for any dry joins or you could reflow them all, time consuming but worth it.
Also what condition are the pins like in the headers on the mpu board?
I did mention earlier, but I hate IDC's as a connector, but it's possible some of the pins have lost their tension and are causing an intermittent join to the header pin?

You could also try reseating or even swapping U10 and U11 on the mpu board, making certain you replace them with the notch on the PIA to the left.

Or before all that, with the game on try wiggling the J1 connector on the mpu board carefully side to side and looking through the rear of the backbox door at the displays you will see if they come on or not?