Let the dogs outside would you?!
The normally closed switch on the coin door is a pain, I have disabled mine by soldering the wires together on one of the switch tabs, although I think the ni-wumpf can be set to ignore it anyway once you can get into settings. Oh, your logic is correct on that.
If wires have been cut behind the coin door it's possible your start button is actually disconnected, check that first. You also need a ball in the outhole to start a game, check that switch as well.
You have manual/wiring diagrams to check coin door stuff?