The power button is connected via a two pin cable.

I would check the clock battery or failing that, pull it altogether.
Then try disconnecting everything from the board except RAM and testing. Don't run it without the fan for too long or it'll fry. I have seen other parts including the fan cause startup failure and other odd symptoms. You might also check the fan for obstructions. I had one where a loose screw got into the fan and jammed it completely. It wouldn't start. Or maybe it was randomly shutting down. I forget.