Should I be hearing game sounds even if the monitors are not working properly?
Yes.
We tend to call that "playing blind".
I wouldn't even think about the monitors until you can get some form of game action going on. The bottom screen showing green means that it is at least "on", but has no true signal coming from the game board.
The top monitor may need investigating later..... AFTER you get game play going on.
If power going to the board seems proper, then I would start by pulling and re-seating all the socketed chips and wire harness connections, including any interconnections between the boards in the cage.