I would check after connector replacement, but as a first test, are you well testing in Vac and not Vdc ? It makes me think it is the case as you measure pin by pin.
In game mode, do you get AC voltage of 6,8Vac on the 3 circuit of j120 (measure between the 6,8 Vac pin and the corresponding return on the connector). Also on J121 for the backbox.
the triacs should be good, there is one per circuit, it would be strange if multiple triacs are out.
The burned connector should be the issue