Also had a look at the drawings and made another conclusion. The only diffence between playing alone and with 2-players is the function of the player unit in connection to the count unit. Playing alone will bypass the player unit. With 2 players the connection will be made by both (with wire R-Y-1), like in 3-players mode ( wire R-G-1). In 4-player mode it bypasses the count unit.
So measure the connections, put the player unit manually to last position (bypass the coin unit) and I hope you find you fault.

Hi Mikola, Chrisbee,
Had a look at your pinball sequence and these are my remarks:

"Ok HF, I put in 4 players, advanced the player unit to the 4th player, simulated ball play, then drain.
It stayed on player 4, but advanced the ball count to 2."
The path to the Ball counter unit step up works nice, but the path to the player reset relay must be activated by the E.O.S switch of the ball count unit. So because of the error here, there must be something wrong with the E.O.S. switch or the player reset relay or its contacts. (or the timing of contact 1C and 2C, 1C must be on before 2C )

"Simulated ball play again, then drained. Advanced to player 1 and advanced ball count to 3."
Allmost good, there must be a connection through player unit and coin unit to the ball count stepup which is fault.

"Again, advanced to player 2 ball count 3."
Good!!!

"Again, advance to player 3 ball count 4."
Allmost good.

"Again, advanced to player 4 ball count 5."
Allmost good.

"Again, game over."
Works perfect!!!

So have look at the EOS switch, the player reset re and his contacts and.. the timing of the 1C and 2C ( 1C must be on before 2C).
The last item you can inspect by looking at relays 'ball count unit stepup' and 'player unit stepup'.