Also if you haven't, clean the brass contacts and wiper fingers on the bakelite disc on the player unit in the head, and make sure the wires have not broken off the wiper fingers. Any 'iffy' connections on those contacts can cause the player unit to miss one or more steps between players or ball in play. Was happening just as described on my Fast Draw and turned out I'd neglected cleaning/oiling that disc, was dirty and dry as a bone.

The coin unit steps up only after a game is started and additional player(s) are added with the credit button, resets at next game start. Also the ball count not advancing when a player scores no points is totally normal. It won't advance if the First Ball relay has not been triggered by scoring points.

Correction on First Ball relay - only applies to the first ball, player one, so should not be staying on same ball after that.

Wait a sec.... "on a 4 player game (and only on 4 player games), the ball will advance from 1 to 2 when it gets to player 4. Then it stays on ball 2 for player 1-3, then advances to ball 3 when it gets to 4." Okay are you sure the player disc is aligned correctly because that sounds like it is one position off. The player position is determined by the inner cog and switches but the ball in play lights are on the attached disc and they have to be in sync.

How about the Add Player Unit relay? All the score motor pulses for the Player Unit have to go thru various switches in that relay depending on how many players active. Clean and adj. as necessary.

On the Player Unit relay scenario I'm picturing maybe one of its switches being 'iffy', then say one of the three 'extra' pulses needed to advance to next ball on a 1 player game or 2 player game might be the one missing. But a 4 player game works okay because it's only doing one pulse per ball serve and that particular switch is fine ... something like that. Sort of the same idea as the player unit contacts being iffy.
yytttytu (resized).png…………… NO wait, P2/3/4G, I think those are in the add player unit itself(?) Same applies though as to a missing pulse.

On my similar 4-player Gottlieb I can duplicate it playing a one-player game that errantly stops on player 4 instead of one more step to next ball of player 1 - if I block the very top switch on the farthest stack to the right in your photo (last stack to rear of unit) so might try clean/adj. of that one. Correction, the last TALL stack.

Also where you said there was one game where the "first ball stayed on ball 1 for the second ball", I'm guessing that was just a legitimate First Ball relay thing with no scoring since otherwise you don't seem to have a problem with the pl unit not even firing once on a one player game.

On my Fast Draw those switches that enable the player unit advances at player 1 thru 3 positions are the top switches on the three tall switch stacks starting from the rear. You can see the stacks activating at player score positions 1 thru 4. I think the single advance needed when playing a 4 player game is enabled by a separate score motor switch. Not sure you would need to take the unit apart but you can remove the two screws and a cotter pin there on it which which lets you drop the unit open and gives you a lot better view and access to the switches.

The two motor switches could be involved but then it would be more random. Unless the motor 2C was misgapped, dirty etc, then it would tend to fail for the one pulse it always gives no matter how many players. I would test it out from playing four player games, then three etc. until you see the missing/screwy firing, then you know which of the motor 2C or the three player unit switches could be the problem, or otherwise which circuit in the coin unit or elsewhere.

Fast-Draw-sidertrty-art-small (resized).jpg