That could be the power hold relay as there are only a couple of wires running to it. By you making it close, the machine is getting power to the various circuits. Once you fix the start issue/s, that should come on automatically But if you are shorting the coin drop switch in the door, the machine will continually just keep firing away. That coin door switch should open and close the once just as if you've dropped a coin in.
Out of interest, did the score reels reset to zero as they form part of your start circuit.
I would advise you to wait until your schematic arrives, get the coin door wiring checked and rectified before you go further. The classic mistake is to try tweaking things and likely adding to your problems. One step at a time is a safe path to walk when you aren't that familiar with EMs.
Lastly, invest in a new mains power lead and run an earth wire to the coin door, and the frame that holds the lock-down bar, and flipper link rods. Oh, and check that the insulating 'fish-paper' on the coin door start switch is sound or you run the risk of a mains voltage shock. Personally, I replace it with a new one and also remove one thin spacer from the switch stack to allow for the insertion of a piece of flexible plastic to the same size of the switch blades with 2 holes drilled to go through the spacer fixing pins so that there is no possibility, once reassembled, of a shock should that fish paper ever fail. Just my preference, as my machines are on 240v mains.