Ideally, I would want to rule out something...probably not likely that both boards are bad, so swapping out the sound board or trying the sound board in another game should help identify which board is troublesome. Unless it is a data cable cable problem. Floating the sound board is not likely to help because this isn't noise along with sounds, but the absence of sounds. You have no music, and no sound effects, correct? The ability to turn up the background noise with the game pot indicates your final amp is responsive right before the speakers, but it appears that a little further upstream from that the final mix is not present to be amplified. You'll probably need an oscilloscope to troubleshoot this.

I would definitely swap the cable, as they are notorious for failing. Worst case if it doesn't fix it, you'll end up with a spare.
Chips can fail for many reasons. Severely overheating ROMs could be a problem. Start with the basics as far as they are concerned. What's your Vcc DC voltage sitting at, condition of socket, and check for activity on the pins. You don't know if the sound ever worked in the games current state: ,Are the ROMs oriented correctly in the board or socket? Could they have been put in backwards at some time, potentially damaging them? If you can't touch them for longer than 2 seconds, I would replace them.

If the aux sound board has a catastrophic failue, you won't get anything because *most* of the sound circuitry is there. You should hear a *bong* sound on boot up if the board is basically fuctioning.