Board is fine just need to do as @thelaw said new connector and header. If you own a dataeast game you will have to do this on every gi harness unless it was replace by prior owner. These connectors were under rated for the load and now time has taken its toll on pin and connector surface causing it heat up and burn up. Also check your power supply connectors too those can be a problem also. I see a wire nut in that 2nd pic also

looking at the pic i bet they ran the red spliced wire to back side of board to fix gi works but not proper repair. As to the other connector looks like you get to do some more header connector work pinball life has the idc tool you will need did it the hard way for to long and bought the right tools goes way faster and less fussing. desolder the header from board them you can work at getting wires out of header connector easier. Also check the back side of ppb board for cracked joints on all header pins replace any discolored headers or connectors otherwise you will be back changing that next time.