Intermittent issues are (almost) never a MOSFET problem.
Given the state of the original CPU (with the lid / heat spreader removed), I can't help but suspect lots of force was used at some point to remove the heatsink from the CPU. Did someone use thermal epoxy on that? If so, I wouldn't be surprised. See if you can use a sharp flat heat screw driver to separate the heat spreader from the heatsink. If you can't, then that explains why it's stuck on there... and again, probably at some point, a lot of force was used to remove that CPU from the motherboard without the latch being open. With this comes the risk of damaging both the CPU socket itself and the BGA under the CPU socket - something which I have done myself once as well (on a LGA775 mobo). Flex it enough times while installing the cooler/heatsink, and that's most likely to be the issue here. Of course, it could also be bad contacts in the RAM slots or the CPU socket itself too... though at this point, these should have cleared when you tested different CPUs and RAM. So I suspect these are not the issue. It could also be dirt/dust buildup causing issues... though again, I think this is less likely to be it. In any case, it probably won't hurt to have the motherboard washed with water and dish soap. Then let it dry for a few days under the sun or in a breezy place. But if that doesn't help, then I highly suspect the CPU BGA is damaged somewhere. The stock Intel heatsink retention mechanism for P4 is already bad enough (at least if it uses the cooler design with double-latches with lobes on each side) as many of these tend to warp the board under the CPU socket. Most s478 boards that became intermittent and died "for not apparent reason" were a case of this, IME, bad caps excluded of course.
If nothing seems to work, remove the CPU heatsink plastic retainer from the board, try installing another CPU, and put any heatsink on it that fits (can be LGA775 heatsink, socket 754/939/AM2/AM3, etc.) without clamping it down. Now try powering on the board. Can you get it to boot? If not, how about if you manually apply some pressure with your hand on the heatsink and then try to power on the PC while still applying pressure? Can you get the board to boot then? If yes, you most certainly have a bad BGA in the CPU socket. If not, then... well, there's still no guarantee that's not the issue. Sometimes a bad BGA joint may not get restored simply by applying pressure. But with 1 out of 3 boards, I find that it does, so why I recommend it as something to try.
As for bad MOSFETs... again, that just can't be it, really. Bad MOSFET usually means shorted MOSFET. Most MOSFETs tend to either switch (chop) or linearly regulate a high(er) voltage rail down to a lower voltage. In the case of the CPU MOSFETs, you get 12V as input and 1.5V as output to the CPU core. If an upper MOSFET was bad in the CPU VRM, you'd get 12V going to the CPU core, and that usually a) causes an over-load / short-circuit protection on the PSU to kick in and b) kills the CPU (though P4's tend to be pretty tough and hard to kill IME, so point b is less likely here.) And if it was a lower MOSFET shorted, you'll have a straight short-circuit to ground on CPU core... so again, a bad MOSFET in the CPU VRM is not possible here given the symptoms you describe. A short MOSFET elsewhere is also unlikely. For example, the RAM VDIMM voltage (2.5V for DDR, 3.3V for SDR): for DDR, the 2.5 or 2.6V supply is typically generated by a linear regulator from the 3.3V rail on the PSU. If the MOSFET that does this goes bad, it will send full 3.3V to the DDR RAM module... making the chips overheat and/or smoke. That didn't happen in your case, so you can be sure this was not the issue. Same goes for SDR RAM: it takes 3.3V for the ICs, which is generated by a linear regulator from the 5V rail typically.