The startup relay was slightly out of alignment. A video shows correct operation.

Did some diagnosing and noticed a number of loose wires and bent leaf switches unable to be actuated. Our current assumption is that the four gears under the play field are trying to cycle to all zeroes on startup. The gears are sometimes too rigid to turn and the contact points that cause the next gears to rollover are likely not clean and getting actuated. Since the machine can’t recognize all the wheels are set to zero, it triggers and cycles everything until it reaches the default (inaccessible) state