That transformer looks fine, very commonly used transformer.
The daisy chain of black wires do go to each microswitch, that should lead back to the JAMMA connector, you will have some extra connectors, just cut them off.
For the player two side you will have to take one of the other black wires on the harness and use the connectors you cut off and join it to that wire, then run them to your microswitches.
Obviously you will have to cut and join the joystick wires to the joystick harness
I can't quite see from the photos, but it looks like the joystick has spade terminals you can connect to, you may be able to just connect your JAMMA harness connectors directly to these instead of using the joystick harness.
I would recommend that, because then if you have to change the joystick, you don't need one with that harness connector.