Make sure your little coin switches are not stuck on.

The ball player unit should be incremented before the ball is kicked out. what happens if you manually increment the player unit 4 clicks while the ball is in the drain? I think the score motor should do that before kicking out the ball if you are playing a game with only one player.

After watching the video, I'm betting on one of the score reel zero switches being shorted closed when they should be open. I'm assuming the relay you pulled open to start the game was the reset relay. There are 16 to check and they could be shorted in the stack.

You just need to verify the switches that pin-it has highlighted above. It's going to turn out to be a really simple and obvious problem once you spot it. That's how it always goes.

Obviously the reset relay is not unlatching at the end of the start-up sequence. This must be because of a shorted switch.

And after seeing the video, I can see it is not the actual coin switch or the reset relay would snap back closed right away.