The control unit is a clutched affair. The entire thing is driven by a motor, but there are separate sections. Review the manual page I pointed out.
Credits vs payouts... there is not such a thing as a payout in a bingo. It's all credits. The deal is that there's a knock off circuit. Other games (older) had a separate button to knock credits off. In a bingo, when powered off and back on, the replay register zeroes. If the game was modified aftermarket to pay out replays directly, that was wired based on how the operator chose to do so. Some did it on accumulation, some did it on reset.
This happens through the switches on the anti-cheat relay, the replay reset relay, and the replay register. You may think the replay register switches are good, but they are a little finicky. A bingo is a mixture of different styles of switches - certain aspects require more precision than others. The replay register SPDT switch is one such.
If it is fine, then there's an issue getting the control unit to shut off, or for certain sets of cams to stop rotating. Gummy clutches will do that to you.
To fix, the control unit must be disassembled and the clutches cleaned and re-oiled or replaced with fresh clutches newly oiled. The oil is neatsfoot compound (NOT oil). This is specified on the schematic. When I clean them, I quickly douse in 91% isopropyl, then scrub clean with qtips/a paper towel, then quickly submerge in a neatsfoot compound bath. The clutch stays in there for about 5 minutes, then take out, pat dry, then reassemble. Each clutched section comes off in one piece after unbolting from the next in the sequence, if you go that route (sounds like you'll need to).
While it's apart, ensure the shaft is clean and deburred. The switches on top of the control unit will all come off in one piece. There's a screw under HIGH tension on the edge of the unit (called the search disc wipers). Be careful when loosening this screw as it will rocket off if you're not paying attention. Otherwise it's pretty straightforward, but a more advanced EM repair technique compared to just adjusting some switches.
Regarding the coin unit - Mystic gate is one of a handful of 40 coin max games. So the maximum amount of replays or coins played per game is 40. My guess is that the control unit cams are not in the correct spots. There are two coils mounted underneath the control unit on the left and middle - these gate various sections. One armature is probably not in the detent correctly? Total guess.
Have you verified/cleaned the coin unit? You mentioned that all the steppers are stepping and resetting correctly. When the game is turned on, it will try to zero credits. Once it verifies that the register is empty, the replay reset relay will let go, then enable coin entry. Putting in a coin will reset all steppers and pulse the coin unit (up to five times) based on if configured for multi-play or single play. Multi-play in this instance is a way to give multiple credits for one coin. For example, a quarter is 5x a nickel, so if configured for a quarter per play, players used to paying a nickel per game would feel better about the change in coinage. You've already tried unplugging the coin door. Is it still unplugged? That would rule out weird issues with the coin switch when tied to the multi-play relay, but I'm guessing that's not the issue.
The manual and schematic are quite detailed - be sure to investigate the areas I mention. All switch positions and their uses are documented.