as BCB said, if you have credits on the register, the circuit to the coin switch and multi-play relay is disconnected.
the main reason is the game circuitry on bingos always decremented the replay register. Players get a little nasty if they stick in a coin and lose a credit at the same time. 'course, part of the circuit is to reject a coin if you have credits, but usually operators disconnected the "coin lockout coil/magnet" behind the coin mech. It tended to buzz loudly. Dixie is worst case when you have credits ... it keeps the coin and does nothing if the operator did that hack
assuming you have no credits on the replay register, then manually closing the multi-play relay should power the start relay via the path through the coin switch on wire 53-8.
the main problem is if the coin switch worked the first time to reset the game, then there's no logical reason it shouldn't work again assuming no credits got added. If that's the behavior you are seeing, it's likely someone has rewired the game incorrectly. If you don't have a three terminal coin switch, the game is definitely rewired.
the normal sequence is coin switch goes down and powers the multi-play relay (which holds itself powered), then when the coin switch goes back up the start relay is powered thru a multi-play relay switch.
the one gotcha is if you hold down the coin switch too long, the tilt trip relay will trip. That's to help prevent people from fishing a coin on a string.