There are two circuits that can send power to the game relay coil. And, yes, one of those circuits that include the Tilt Latch Relay, Start Trip Relay and Game Over Relay, all have closed switches in the circuit. This would mean the game relay coil would be always energized at the start of the game. Perhaps there is a three position switch in the circuit or there is a mistake on the schematic???
The other circuit that includes the label "to 36-2" might be a source. This circuit goes to the score relays, which would send power to the game relay coil when a point was scored on the playfield. I would check the following switches on the score relays and match unit:
1 pt score relay switch - yellow/brown wire and green/white wire
10 pt score relay switch - yellow/brown wire and grey/white wire
100 pt score relay switch - yellow/brown wire and black/white wire
10 pt score relay switch - yellow/brown wire and yellow/black wire
100 pt score relay switch - yellow/brown wire and yellow/black wire
Match step up EOS switch - yellow/brown wire and green/yellow wire
Make sure these switches are clean and adjusted properly. Look for shorting wires at the switch stacks and switch stack lugs.
Another related "gateway" switch in this circuit is the outhole index relay switch. Check if there is a short at this switch - shorting wires at the switch stacks and switch stack lugs. You can measure voltage at each switch blade - yellow/brown wire and black/white wire.