Welcome.
The transformers in these games are tanks.

With the mains fuse removed, how are you managing to power on the game and measure correct voltages?

Measure the AC voltage across points E11 and E12 on the rectifier board (one meter lead on E11, the other meter lead on E12). How much AC voltage do you measure? It should be about 12 volts AC. This then goes into the BR2 bridge rectifier to be converted to the 11.9V DC.

When you measured that 9.91V on the rectifier board, was the solenoid driver board connected?

E11 goes to the trace connecting to the bottom of the F3 fuse. E12 goes to the trace at the top right pin of bridge BR2.

See what barakandl said about capacitor C23.

Sorry, what's a PDB?

First, with the machine off:
Disconnect the J4 connector from the MPU board
Disconnect the J1 connector from the sound board.
Both these connectors carry the 11.9V to these boards.

Power on. Does TP3 on the rectifier board now measure 11.9V again?

If not, with power off, disconnect capacitor C23 on the solenoid driver board, then power on and remeasure.
If the voltage now returns to 11.9V, then C23 is suspect.
If not, you might be looking at a faulty 5V regulator on the solenoid driver board or something downstream from it that's overloading the regulator.

You mean capacitor C13? It's probably not going to be causing your sound cutting out problem but if it's bad, it's very likely its brother of same value at C5 is bad and that one interfaces sound from the pre-amp to the output amplifier. Replace all the electrolytics and see how you go.
Another thing, since the problem is thermal, you can use freeze spray (or air duster in a can which expels refrigerant) to cool components to isolate the problem.

It's probably the 4543 BCD decoder chip on the display board. It's responsible for generating the numeric characters. To display a "F" like character with no top segment for every digit would otherwise require failure of many other components. Suspect the decoder chip first.

Does the sound go quiet instantly or does it fade to quiet over seconds/minutes? If it's slowly you're looking at an issue in the amplifier section.

Have you freeze sprayed U8 (mixer/pre-amp) and U9 (output amplifier) - in terms of chips, they're the first suspects.

When it's gone quiet, what voltages do you measure on the outputs of the amplifiers? The schematic says output pin 4 of U8 should be around 2.2 volts and the output pin 4 of U9 should be around 6 volts.

Also check the voltage at TP1 on the sound board which should be around 13 - 14 volts.

When the sound has died, what happens when you press the red test button on the sound board? Does it play any sound effects?

Yes, U9 on the heatsink is a TDA2002 which is an audio output amplifier.