Welcome to Pinside. Here's the relevant part of the schematic:
Gay90sBallCount (resized).jpg
This circuit comes up regularly in this forum so you'll find more detailed explanations in other topics. (Look for Player Unit or Coin Unit.)

The short answer is that in a 1 player game the path in red should fire the Ball Count Step Up (S.U.) coil to advance to the next ball when the ball drains. What you're seeing instead is that the Player Unit Step Up coil (red box on the right) is firing to advance to the next player. It's likely that either the Score Motor 1B switch in the red path isn't closing, or the path through the Coin Unit, which is shown in the 1 player or reset position, isn't complete.

If those two are working properly the Ball Count Step Up coil would fire to advance the Ball Count Unit, and the EOS (End of Stroke) switch in the bottom left corner would fire the Player Reset relay which would flip the Make/Break switch in the middle red box to fire the Player Unit Reset coil instead of the Player Unit Step Up coil.

/Mark