How have you tested the power supply? It may measure 12V when removed from the Mac, but it could have issues operating under load. If you can, probe the power supply outputs with it connected to the logic board, and see if anything happens when you attempt to turn on the computer.
Another thing to check is if the fans spin at all, a "quarter spin" behavior when the power button is pressed can be indicative of issues with the logic board that would require professional repair.
Finally, are you sure there isn't an issue with the power button itself? If there's absolutely no activity when the button is pressed, maybe it's defective. You may be able to trick the mini into powering on by shorting out the power button's pins where it connects to the logic board, but you must be careful not to short anything else out inadvertently.