Alright, maybe I'll go ahead and have a look. First, the Delay Relay should not be pulled in at start-up. I know you and Rolf were discussing it coming on when there was a slam tilt. That relay should be off in any other situation except a slam tilt or a kick off. There is a normally closed switch on it that basically provides all the grounding for all the sections of the coin up and start process, so if the delay is on none of it works. I think you've troubleshot that pretty well and it's probably not the issue, but just to be sure look at the Delay Relay and check the switch with the black-red and black wires coming to it. It should be closed when the relay is off and open when it's energized and pulled in (on).
So besides the Delay Relay, the chain of logic to getting things going seems to be this:
A make break switch at 1F on the score motor should be closed (probably is, I wouldn't tinker with the score motor until you absolutely have to), there is a normally closed switch on the Reset Relay that should be closed (has a black-white wire to it), the credit (start) button needs to be pressed, the Credit Unit Zero switch, which is open when there are no credits on the reel, should be closed, and finally either the Game Over Trip (the one on the Reset Bank?) or the Game Relay (Game Interlock Relay?) need to be closed. If all those things are closed it creates a path to the Credit Relay, which will then hold itself on. The Credit Relay activates the Coin Relay, which activates the Reset Relay, which essentially gets the game going.
What's weird is that it sounds like you weren't anywhere near any of these things when the problem started. Weren't you adjusting settings in the backbox? I'd triple check the credit unit, since that's the only thing that would have been close by that's relevant. Tug the wires, look for bad solder joints, make sure the free play switches you adjusted are the Credit Unit Zero switches (grey-yellow and white-orange wires to that set). Be absolutely sure the credit unit is really telling the game that it has credits on it. Is the credit light on, on the bottom left side of the apron?
It also looks like the Coin Relay may be holding the Reset Motor on. The Coin Relay holds itself on through a switch on its own stack and a normally open switch on the Game Relay (whichever one that is, the one on the Reset Bank or the interlock?). So if the Game Relay doesn't drop out, the Coin Relay stays energized and keeps running the Reset Motor forever.