The fact that it was working when you bought it and not when you got it home screams "bad connections". And as you have already mentioned, system 80 games are known for having poorly constructed harness/pins/connectors. One of the easiest/quickest/cheapest steps you can take to make these connections better is to clean the finger board contacts (edge of pcb where connector attaches) with a fiber glass eraser. This is constantly an issue as they become tarnished, resistance builds up and signals are lost or floating. Don't use sand paper, don't use an eraser, get a fiber glass eraser and clean them properly.
From there you want to remove and physically look at each connectors pins and determine if they are tarnished, bent down, broken or loose. Pins that are not inline with the majority of others are probably worn/bent down and need to be removed, either using a pin remover tool or bobby pin, paper clip, exc. and then cleaned/bent back to their original position so they can make firm contact with the board edge contacts. If they don't feel springy or look rough they may need to be replaced. Specifically the power and ground pins are your main culprits. Most of the other logic signals are low voltage lines that probably have little to no damage on the pins. Confirm continuity with your meter on every pin or connection that is suspect.
Do these steps first before repinning every connector. Pack a lunch if thats your plan as it will take a while.
For your sound board, its good that its outputting in test as that means the amp is good. I don't have time to look at a schematic right now but I'm curious if the input signals during a game are not getting through to the board. Maybe a bad buffer on the CPU?
For your drop targets, do they ever reset? During the start of a game? Or are they down 100% of the time? If down all the time check your reset coil for both power, ohm, and leads. Make sure the leads from the windings or the wire harness leads are not broke off the lugs. If all good then confirm/test associated driver board power and pre driver transistor.