I have a 73 gottlieb so going off that I can't even see any connection to the outhole and the start relay.

Just want to make sure when you say it's reseting you mean the S relay pulls in and the player unit cycles around to reset the reels and go to ball one, player one?

Maybe somewhere in the coin door there's some mis-aligned fish paper or dampening blade that's shorting a switch somewhere like this? http://www.pinrepair.com/em/gtbswbad.jpg

Have you tired making sure the coin door itself is grounded and nothing is shorting against it? I spent 3 days chasing an issue before where coils were activating because switches were shorting up against the metal frame.

Other then that the only thing I can think of is check the S relay itself, though considering you said unplugging the coin door fixes the problem I don't think that's the issue. Does everything still work properly if you unplug the coin door, and manually start a game by actuating S with your finger?

Cleaning the plugs is always a good idea. I find either a thick brush or a 600 grit sandpaper works the best.

Also clean the female ends. I use pipe cleaners and a little bit of 99% isopropyl alcohol to clean the female ends. Afterwards I just blow out any remaining gunk.