Hi jfoust
I wonder if Your pin makes the Game-Over-Relay to trip when You toggle-on the pin. See the JPG --- toggle-off the pin and unplug the main power cord (Safety Reasons). Do manually fumble on the Game-Over-Relay - make it latch - make it trip - do this several times - are both states "stable" ? Finally do latch the Relay. Then plug-in, toggle-on and watch the relay - DOES it trip ?
AAA - Yes it trips: Then my assumption of "fault causing Your problems" is wrong.
BBB - No, it does not trip: Toggle-off, unplug main power cord - YOU do manually trip the relay --- then plug-in, toggle-on, start a new game --- does the pin shows nice reaction - starting ? Greetings Rolf