My first question, does the machine reset at the beginning of each game?
The score motor is started by one of many relays SWs, then it keeps running until it completes at least one cycle (180 degrees). Normally the SW that keeps the score motor running, in located on the top of stack on the cam nearest the motor. Red/Yellow and White wire.
If the score motor is not making it back to zero position, I would start by checking the connection on the above SW. Odd that propping one end resolves your issue.
If it all checks out electrically, then the gearbox is a good option.
