So I cleaned up some solder points and reseated chips... The sound is steady now but still very low. Also I've noticed that on startup sometimes it won't register the 1P or 2P buttons (so I can't start a game). After fiddling a while (rebooting a bunch of times) it seems to work for a while but then comes back...
Looks like there may be two separate problems going on here. The sound is likely unrelated to the button problems.
I was taking a closer look at the picture of your AR. I'm not too sure about the solder work on it. It looks like whoever replaced the diodes, didn't heat up the through hole enough to let the solder flow all the way through. It makes me wonder what the solder side looks like.
It sounds like you are getting audio from the game board. It is just not coming out the speakers at the correct volume and is being distorted. As far as I know, there is no easy solution to figure out the problem. I can see four potential causes. Starting from the output and working backwards,
1) blown speaker
2) faulty volume potentiometer
3) bad TDA2002's on the AR
4) bad LM324's on the game board.
It would be easy enough to test the speakers by temporarily swapping out another speakers (I don't think this is the problem). You stated that the volume pot doesn't change the messed up sound when it is adjusted (I don't think this is the problem either). You said that there was extensive work done the AR. This, and the questionable solder work, leads me to believe this is the most likely a problem. How are you with your soldering skills?
Maybe someone else on here has a different opinion.