J2 on power supply to J1 on CPU board -- that is output from power supply and the absence of that cable won't affect LED's.
Check to see if you have good power coming into the power supply. Always make sure you have good power at TP4 BEFORE installing power J2 to CPU board cable.

LV input -- Measure voltage between E4 and TP3. Should be an unregulated 12V (ballparked at 12V).
LV output -- if LV power in is good, measure voltage between TP4 and TP3. Should be regulated 5V (roughly within a half volt). Once you have a voltage at TP4, you can bump the voltage up or down with the potentiometer. Be careful doing this - going too high will trigger the crowbar circuit (SCR1). Crowbar circuit is an over voltage circuit that protects the CPU circuitry, it kills the 5V if it detects too high of a voltage. Only way to clear the crowbar is to shut machine off, turn pot back down and then turn on again. Once you get that going, dial in to about 5.05V.

HV input -- measure voltage between E1 and TP3. Should be unregulated at about 76V.
HV output (larger displays) -- measure voltage between TP1 and TP3. Should be 60V within about 5%.
HV output (small display) -- measure voltage between TP2 and TP3. Should be 42V (should be voltage measured at TP1 minus 18V).

Is the connector that connects to driver board's J4 tucked down into the machine still?

Most voltages look good except TP1 is off by too far.
Measure voltage between E2 and TP3.

Wiggling connector for things to come back is a sign of a bad connection.
could be bad solder joint, bad crimp connection within plug or broken wire.

Wrong voltage at E2 --
E2 voltage is established by the voltage rating of zener diode CR5. CR5 is 62V +/- 5%. Worst case upper voltage for CR5 = 65.1V.

One of three things here:
1 -- CR5 is bad
2 -- CR5 is wrong value
or
3 -- bad solder joint at CR5.

He may have been damaged...
Or maybe --
can you make sure there is good conductivity (no resistance) between the E2 terminal and the cathode (banded end) of CR5?
And make sure there is good conductivity between CR5 anode (non-banded end) and ground?

OK -- you may need to replace CR5.

The 2nd power supply looks to be running better than the first one, a tad high on the HV side but not bad. You will need to turn up the low voltage 5V potentiometer. Turn slowly to about 5.05 volts.
Does sound like you need to repin the plug. I would repin all three.

If you need to replace the plugs, easiest is to use CS156 style plugs as shown here:
https://www.greatplainselectronics.com/products.asp?cat=81
But these plugs were often the discrete crimp type plugs which means you can easily repin them.
Crimp contact, buy a few to practice with if you aren't familiar with this. Good, general purpose contact for power supply is 08-52-0113. Can also use 08-52-0072 which is often used within edge connectors as well.

And keys -- 15-04-0219. One per connector if you are replacing connectors instead of reusing.

If you need a crimper - 1026-CT is a good, entry level version.
https://www.greatplainselectronics.com/products.asp?cat=80

I also have the W-HT-1919 from waldom.
Some people swear by these... some people swear at them.
https://www.greatplainselectronics.com/products.asp?cat=59

As you progress along, you may find some edge connectors that need repinning. A choice of contacts for those can be found at the top of this page:
https://www.greatplainselectronics.com/products.asp?cat=77
Reusing the IDC plugs is possible but can be a bugger. As the replacements disappear, more people may be stuck using the original IDC plugs.
No replacement contacts for the black, double sided edge connectors. Connect/disconnect those plugs as little as possible.

No kits for the displays, not really anything to do to these. For the Gottlieb VFDs - if they're working, best to leave them alone.

Based on prior conversation - sounds like just the 6-digit displays are not working and the small, 4-digit is working.
Did you measure the grid voltage to make sure it is getting to the displays? And the 60VDC?

So... apparently I am mixing up your conversation with another? When you turn it on - no displays (as you say). But I thought you said the machine would run but no displays? Or was this the other guy that said this? Apparently I'm not multitasking very well this weekend.

Voltages listed in post 22 are good enough to get you going.
Zablon made good recommendations.
I'd send you off to pinwiki to do power mods as mentioned but their site appears to have been hacked.