Ill have a glance at the Schematics tommorrow and make some suggestions.

A completed switch in the ball count unit (still could be an issue) + all the score reels runout switches closed + 4D closed on the score motor will fire the AB Reset coil = Game starts.

The one part that is confirmed is the runout Switches (Done at Rollerball's meet)

The Second part that is confirmed is the Ball count unit is adjusted - BUT - there could be a wire issue - Daniel's suggestion of an alligator clip is a good one

The only part left is the 4D switch on the score motor must be closed. do you know how to find that on the schematic ? The switch is physically the topmost stack ..