Hmm.. there's a word for people like that!!
OK, I guess I would set up the working system, and look at the different voltages and signals around the mysterious A12 component.. It looks like this bit of the board is the part that is connected to the video out at least.

Ideally use an oscilloscope, and see what the "CVBS" signal actually looks like - https://en.wikipedia.org/wiki/Composite_video

Then do the same with the non working board, and try and find the differences.
(I assume the working board has the same IC on it with marking?)

It seems like the CVBS signal should come from the ADV7177 IC - pin 31 It must then go into the damaged board, and pass through part of it and finally out of the connector on the board. At some point the signal gets interrupted on the damage board (or does it). Find where the loss of signal occurs, and then check the component voltages and signals around it and compare with the working board to see where the differences are. Then hopefully you should be able to find where the fault is!

You might also try putting an image of the A12 component on here and ask if anyone knows what it is.. maybe an amplifier IC?

 

I will try and do that. I don't think my boss is going to allow me to spend much time on troubleshooting it too much more. You can buy the whole controller on Ebay for $120. In addition, they are using the machine/production line this week so I'll have to wait til Monday at minimum. I was optimistic about emailing the company. I didn't get to speak to someone in engineering support. Only the sales rep I'm used to dealing with. The chip image is already posted in this thread. I'll see what I can find out and report back.