Boy do I have a story for you today!
I started taking the pentium Pro apart for the recap and when removing the CPU's from their sockets I noticed that each of them was missing a grounding pin in the same place:

I was able to recover both pins and will ask the avionics at work next week if they can maybe solder them back. (I have another Pentium Pro which is in perfect condition that I'll use to test the machine in the mean time of necessary.)



But it still got worse than this!
When I was removing the base board from the machine, I noticed that I couldn't remove the ATX power coupling. On closer inspection I noticed that they had partially melted! (The only way in which I could get the board out of the machine was to cut the power couplings)

The underside of the base board doesn't show any scorch marks, which is a good thing!

I measured the 5V and 12v rails on the power supply under load, and they're fine:


The bad connection on the power coupling's more than likely the problem after all. I wasn't smelling exploding caps, but I was smelling the melting power coupling.
So I went off to the electronics store and purchased a new MOLEX KK 396 12-pin connector to replace the damaged power coupling:

I had to purchase a crimping tool too to fit the female connector to the power supply:

The power supply's rewired:

I'll have the male connector fitted when the boards are re capped.
Here are the specs on the MOLEX KK 396 12-pin connector:

So it can handle 35 watts per pin @ 5V, and there are four 5V pins, so that's 140W, which should be enough!
Here's a pic of someone else's board and the same pins are burned! (These things should have been fitted with 24-pin power couplings!!!)

@legodude on SGUG Discord suggested that I add some extra MOLEX power to the underside of the base board:

This is probably a good idea!