Do you have any more info to share how to "wire this up like a normal arcade joystick"?
No to be dismissive, but that's a bit like me saying "just drive to the store" and then you saying "can you share info on how to drive to the store?"
If you google around you will literally find thousands of results on how this works but here are the cliff notes:
Buttons have "micro-switches" which have 3 electrical prongs on them.
Common (or COM)
Normally Open (or NO)
Normally Closed (or NC)
imagine this is 3 wires and when the microswitch is NOT pressed COM is connected to NC, and disconnected from NO (hence where they get their names). Then when the switch is pressed COM is disconnected from NC and instead connected to NO.
Buttons have 1 micro switch that is pressed when you press the button.
most Joysticks have 4 microswitches, one for each direction: up, down, left and right , this includes VOOT sticks.
Now typically for buttons and Sticks NC is not used and left disconnected. Sometimes the microswitch on joysticks and buttons wont even have an NC prong.
Typically Ground is connected to every COM pin (such that all the COM pins are wired together) and then the NO pin of each button and each joystick direction is wired to a specific pin on the interface device (in this case the control panel connector diagram you found earlier). Some buttons with only 2 pins wont have them labeled because with only 2 pins it doesn't matter which is witch.
Hopefully that answers your question.