R1 is specified as a 600 ohm resistor (check the schematics). They are however harder to find so your boards that have 680 ohm resistors have had it changed (commonly accepted replacement).
This R1 resistor however is on the solenoid 43V power rail - nothing to do with the 12V & 5V lines.

Also see this current thread with a similar problem that seems to be caused by a suspect U14 - barakandl also mentioned U14 above.

https://pinside.com/pinball/forum/topic/bally-mr-mrs-pacman

The OP in that thread used an icecube in a watertight plastic bag to freeze components and track down the issue (incase you don't have freeze spray).

When you get a solid lit LED on a warm power-up, what happens if you then do a manual reset of the MPU board - i.e. very briefly short the reset line at the CPU (U9) to ground? i.e. just quickly short pin 40 to pin 39 on the CPU.

Yes sounds like you have a problem in the reset circuit. You can add the capacitor as per Clays guide but that's only masking a problem.

What's the voltage across zener diode VR1 (in the reset section) on the MPU board?
Have you run this board on the bench to see if U7 still gets hot? When you say hot, can or can't you leave your finger on it?

Your picture shows a Fairchild brand 6810 at U7 - are the other 6810 you've used also Fairchilds? I've just put a Stern MPU-100 board on the bench with a Fairchild 6810. After 10 minutes the chip gets warm (in free air). It's certainly not too hot to touch, but it's not cool either.
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Not really. BTW, what happens if you put this board into one of your other Bally machines?

Can you also amuse me and remove the coin battery. Does the board still fail with a locked LED when warm?

This might confuse matters, but have you done @barakandl's recommended changes to the reset circuit after installing the NVRAM which from memory is to remove R11 (the 82 ohm 1W resistor that gets hot) and replace diodes CR5 and CR7 with a jumper links?

Ah sorry, I went off ancient memory of Clays guide. Just looked and the addition of the 470uF is more a suggestion to test delaying the reset. But yeah, 1uF is more like a reasonable value.

The voltage across the 8.2V zener diode in that circuit controls the short reset hold time on power-up based on the slightly longer time for the 12V line to reach supply rating.
If the voltage across this zener is less than 7.8V then I'd change it. If it's close/just under 7.8V then for test, you can try temporarily adding an extra diode in series with it which will add another 0.6V to the release threshold of the reset.
Also test resistors R112, R1, R3 and R2 to make sure they're in spec.