Actually, I'd just remove the side panel, trace the wiring coming from the power, sleep, sound, usb on the front panel to the motherboard, note where EACH of them connects to, and unplug them. Then using a flat screwdriver, very carefully touch only the two pins that went to the power on - off button together using the screwdriver. Be certain that you are shorting out the correct two pins. Only touch the two that go to the wiring that connects to the power on/off button. If the system exhibits the same issues as it did with the front I/O and power connections plugged in, then that isn't the problem. It's an unlikely possibility, but I've seen it a handful of times over the last few years so worth checking. Those wires can fail or become shorted, or break. The front I/O mini boards (On models that HAVE a mini board up there for I/O) can fail. Power button can fail. Sleep button can fail, and I've seen a faulty sleep button switch try to constantly put the system to sleep over and over again.
Again, it's not a high probability solution, but it's possible.
If that's not the problem, then disconnect the drives to see if one of them is shorting out or faulty. Triple check that the CPU EPS and ATX power connections to the motherboard are FULLY seated, and even unplug and reseat them. Same with the memory and graphics card. Sometimes all it takes is removing something and putting it back to fix the problem. Often in fact.
Power supply IS the most likely suspect though, especially since you've swapped out the majority of core parts and still have the same issue. Only things that are common to both builds are PSU, graphics card, storage devices and case, so eliminate them until you find the problem.