Step 1: find the player unit step up coil on the schematics. It's around column 6-7.
To make any coil fire, you need voltage to reach both sides of it. Looking at that coil, we can see one side is attached directly to the thick horizontal 80 wire, and the other end goes through some stuff to the 30 wire. So to fire it with clips, you just need to bypass all those switches and hook it straight to the 30 wire.
Similarly, for it to fire normally, all those switches need to be closed to make that same connection. If you make a list of the switches, you'll see there's one on the player reset relay, one on motor drum 4B (the one in a circle), one on the extra ball relay, one on the ball index (playfield valid) and one on the outhole. That all makes sense. If the ball's not in the outhole you don't want to go to the next player. If the playfield is invalid you just kick the ball back out. If the player has an extra ball you stay on them.
To take a brute force approach, you could just examine and clean all of these switches and it'd probably fix your problem, assuming the relays they're on are all firing properly (if the ball index relay was never turning on then adjusting the switch on it isn't going to help and you'd have to go look at that). You can be smarter though and maybe save some time by looking at what else these switches control. you'll note that the player unit reset is on the same line, only the player reset relay switch is different. So if it goes back to player one from player 4 properly, then you know to check the player reset relay switch first, as it's the only one they don't have in common. If the ball count unit step up solenoid to the right works properly, then you know it can't be the extra ball, ball index, or outhole relays.
However, you have two problems. Your ball count is also not stepping up on multiplayer games. Does it not advance even if you manually move the player unit to the last player? If so then the ball count problem is just a symptom of the player unit problem and can be ignored. If not you'll note that it goes through the Coin Unit (which is really the player count), and you might want to make sure that stepper disc is clean and the wipers are making good contact.
As for adjusting the score motor switches I don't have many suggestions. Usually they hinge up so you can get a more direct side on view of the switches. Turn the game off and spin the motor by hand, trying to watch the contacts open and shut. Get a switch adjuster with a 90 degree end to stick in between the stacks. If necessary remove the whole bottom board so you can get your eyes right up next to it.