Test the two rectifying diodes on the power supply (these are the two large diodes on the lower right corner of the supply by the fuse.) Just fixed a Gorgar that was resetting, and one of those two diodes was shorted.
On williams s3-6, they only half wave rectified the AC power with two 4a1 diodes. (opposed to using a bridge rectiifer with full wave rectification.) If one of the diodes shorts, they you only get 1/4 wave rectification! there's no way the 18000 mfd cap can filter that enough, and though the game will boot and run, any flipper action will reset the CPU.
i know you said you changed the power supply, but that's where i would start. After checking those two diodes, i would replace the 18000 mfd filter cap (you can use 15000 mfd, but don't go lower than that.) Also of course check the flipper coils. Tug on the coil diodes and make sure they are not cracked/broken/missing. Also inspect the large square connector on the power supply, and the long .156" molex power supply connector at the top.
By the way, that post about the driver board flipper relay diode/transistor. That would be really weird if that had an effect. Yes the relay does run at 5 volts, but the relay coil is very high ohms, and it seems unlikely that would effect anything here. A more convincing idea may be that the bottom +5/ground connector on the driver board got reseated in the driver board removal, and that had an effect on the game's resets. Anyway, that's how it looks from the cheap seats...