Verified problem. One press usually starts two games.

Inspected switch. Wiring was non-original, with a diode connected on one end to the switch and the other to a wire, and wrapped with electrical tape.

There was also an extra wire. Looking at circuit diagrams, the extra wire belonged to the memory protect switch, which had broken off the coin door switch. I desoldered the extra wire, added a spade terminal, and put it where it belongs. Behavior did not improve.

Verified that the diode was good, but looking at the circuit diagrams and comparing with Gorgar, it was connected incorrectly. Soldered things back the right way and the behavior improved but sometimes still started 2 games.

Searching on the internet, a few people said that the problem could be the switch gap, such that if it was too large, the connection might bounce. Narrowed the switch gap and moved things closer to the button, and also used the flexible file to clean the contact points. That appeared to fix it; it worked 16 times in a row.