If it is an obscure logic failure, then you most likely won't find it. It would be buried deep in the thousands of little lines on the logic board, impossible to detect with the human eye.
My experience with Macs is that if the GPU fails then it would turn on, but have a distorted or blank display. The GPU failing should not affect the battery light (unless something is REALLY bad and short circuiting something.) An iMac of mine had a failing/failed GPU, and it's display just was looking a little whack for a while before the GPU died completely, and then the only color the GPU would render was pink with ugly white lines all over the screen.
Replace the DC-in board, as that is definitely a thing that could cause this problem. Reseat RAM, and try reseating the connectors that connect the DC board to the logic board.