Well, I've never seen the inside of a Mini so bear with me. But is there an internal PW switch, or is it a wired one? I can't tell you how many times I've freaked with my PowerMac 500 before realizing I forgot to connect the front panel.
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.
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