Cz7asm,
A couple things that kind of don't mesh with this logic that I'll bring up.
Firstly the fuel is not based on the ATX standard, it's based on the WTX specification that started up and failed to take hold.
https://phantom.sannata.org/download/file.php?id=30455
So your assumption of a 5 V only standby line is incorrect. While I didn't find the main PSU document specification the above file states clearly in the last section that the WTX standard provides both 5V and 3.3V standby for the system.
I know you claimed that the power supply you're using works fine another system. I would still feel better if you had another power supply, even if it was an ATX power supply using the established adapter board to confirm that versus one power supply that might be suspect.
To me this sounds like a power problem. Now I think you should attempt to use a multimeter that has a mini and max feature and see if you can probe the back of ATX connector while it's on the board and cycled the power to the power supply using a power strip or the power cord on the wall and see what Your meter says. My thought is that you're just barely getting the correct voltage for things to work.
3.3 V requirements are actually much more common than a 5 V requirement for surface mount components that are also semiconductors. So I'd like to believe that the trick of constantly unplugging and re-plugging the motherboard load using the 24 pin connector from the power supply is in fact resetting the power supply Over and over again until it produces the correct voltage.
Now there is the possibility that you have a failing semiconductor in the power path on the mainboard that is sometimes sticking in the on or off position that is causing what you might consider to be a short circuit on the motherboards power system that appears to be a load that isn't normally there at that stage and potentially that's pulling the voltage lines a little too far down.
If you want to compare the boards with the problem to the board that seems to work with the same power supply what I would advise you to do then is to get a good quality multimeter and use both ohms and diode mode tests on every single pin on that 24 pin connector to main ground through the connector. The multi-pin connector that's much smaller is the 12 V VRM power system. It's important but it's not seemingly something that you're involved with so I can easily say that 12 V power is not your issue here nor is that VRM as it only starts up through auto power on second stage. So you're squarely looking at 5 V and 3.3 V in that main 24 pin connector. Go ahead and make a simple listing of the pins and the diode mode reading you get snd the ohms mode readings you get. To be thorough you need to reverse the polarity for another round of testing.
That is to say you need to first do multimeter black common probe to ground and red lead to the component under test on each of the 24 pins that isn't ground. Then start another list and reverse the leads with the red probe being on ground and your black probe being used for probing. This reverse polarity should often come up with a different measurement. Sometimes it'll come up with appears to be a direct short and those are normally protection diodes to ground That are usually fine. You're not totally caring about what the values are only what they are across all three samples.
The point of doing both these tests and the reverse polarity of the same tests on all the boards is looking for differences. Your supposition is there something going on with just the two boards. Making the assumption that it's not the RTC module which is modular and you've already plugged in and out then it must be something not modular on the board. Since the electronics won't come up they're not meeting some form of voltage threshold or current threshold that the system requires to start those components.
Create a listing for all three boards involved. The two that don't work and the one that does. If you find one or more pins that do not measure anywhere near the same. Then you have your culprit for investigation. I would say anything that measures outside of 10% is something you should at least note but I would say it needs to measure 20% outside or more for you to really put your attention on it.
The vast majority of repairs I do are based on a comparative method. I'm not an electrical engineer and I don't have secret schematics that I hide from everyone. I go by comparison most people do this as well. You have a working board and you have a non-working board and as long as you use a multimeter With the board with unpowered there's very little chance you can do any damage except for scratching something with the probe tip. I do not recommend doing much live testing as that's not what I do and one slip up could cause horrific damage. So I recommend boards off and disconnected from the power Supply and removal of both their wiring harnesses. You do not want the power supplies wiring harness to be part of the circuit while you're measuring it. So you want to act like the boards have been removed from the case in that there's no power supply connected.
I think this technique will give you a much greater insight if there is a variance in the board as perceived by the power supplies 24 pin connector. Remember the 24 pin connector is what you're fiddling with so something on that is where the problem is connected to.
My view is still but the power supply is weak but That isn't 100% reality unless you can get another working power supply and prove the same issue. If that's not available then performing both diode forward voltage base measurements and ohm resistance measurements in both polarities on every one of those pins is the only way to tell whether there is a difference or not from the power supply's perspective at that point.
NEED -> 
