I'm pretty sure that the green wire that is daisy-chained from one switch to the other is the safety ground wire, it eventually connects to chassis ground at one end of the green wire. You can follow it over to one side and verify this by seeing if the end has a lug that is screwed to the metal chassis. The green wire doesn't have to be connected to anything for the buttons to work, it is there just in case a short were to happen in the machine and the player might get shocked. This is electrical requirements by UL and a good practice, even though the buttons are plastic so it is hard to imagine how someone would get shocked from the player pushbuttons.
The green wire is connected to a metal post on each button. That green wire is like the 3rd prong that's a little longer than the other 2 prongs on a 120 volt appliance electrical cord, nice to have for safety reasons but not required for things to work. Some people remove the green wire to get it out of the way when working on the buttons, then they never put the green wire back.
There is also a pair of wires connected to each button microswitch. And there is another pair of wires that connect to the button to light the little bulb inside, if the button is the type that has a bulb.
So all in all, you just need 4 wires to the button at most.