The game ending issue sounds like a switch on the score motor. Occasionally the spacers between them fall off and cause trouble. I've had this problem 3 times on single-player Gottlieb games due to said problem, but never on a multiplayer machine yet, so I can't help you there from personal experience.

As for the relay, you can tell you're getting the correct pulse/signal from the score motor as all of the other reels share this same pulse.

Let me just through something weird out there, I had never seen or heard of this issue before I personally had it but it changed how I looked at pinball and coils (people always say "a coil works or it doesn't" - WRONG.), I had a coil in a machine that was perfectly burnt to the brink of dying but not quite to where it would make and break it's internal connection occasionally and work sometimes and not others. It was so bizarre and almost unheard of. It made troubleshooting nearly impossible as the game itself was in-check perfectly fine. Make sure that is not the case with your coil. Is it burnt at all?

Additionally, on these later games the reset is controlled via the AX relay.

There is switches for each score reel on there that needs to be clean and properly adjusted for the reset to work properly.

It is a common problem point for all Gottlieb pinball machines.

I can almost bet your problem lies there. I am surprised it was not mentioned yet. I would assume this kind of electro-mechanical error would effect both players though rather than one, which may be why it wasn't mentioned prior, but who knows.