Clean the fingers and disc on the Ball Count Unit. Post back here with the result.

Check these 3 switches for proper operation, when a ball is on them, and when not.

For a switch to work 3 things are necessary:
1) When open, there should be a small space between the contact points (duh)
2) When closing, the long blade's contact point should push the short blade's contact point enough to move the short blade
3) The contact points should be clean, which they usually will be if #2 is happening

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When the last ball in the trough gets kicked into the shooter lane, the #1 trough switch should open and release the power feed to the ball release relay. It seems like this isn’t happening and the ball release relay is staying pulled in.
Inspect the #1 trough switch for shorted blades and wire connections. Might even be the mounting plate of the ball release solenoid causing the short.

The schematic shows a Y-BR wire connected to the jumper between the 1st and 2nd trough switches, not a white-black wire.
Is this a schematic error or is the correct wire somewhere in the general area possibly hooked up wrong.