Gigabyte Motherboard no-boot/no-POST intermittent issue

The symptom:
All fans (case and CPU) will start spinning at power-on, CPU fan will spin at high-speed and then slow down over a period of about 3 seconds, system will not beep, displays nothing on screen, will not POST, enter BIOS or boot. Pressing the reset button does nothing. I have to hold the (case) power button down for 5 seconds to power-off or use the switch on back of PSU. If the case power button button, the system will sometimes restart by itself.

Specs:
Gigabyte H85M-HD3
Intel Haswell I5-4440 CPU (box)
Kingston 8Gb DDR1600 + Kingston 4Gb DDR1600
Gigabyte Green Power 550W PSU
Cooler Master K380 ATX Mid Tower

I bought a new motherboard, only to find that that did not fix the problem. I replaced the 4G RAM with an 8G DIMM, which also did not resolve the symptom. I've since noticed that unplugging 8-pin CPU power connector to the motherboard, letting it stand for a few minutes and then plugging it back in and powering-on will result in the PC booting normally once, with all subsequent starts failing again with the same symptom.

Is there a way to easily troubleshoot the PSU and CPU to isolate the problem ?

Another topic describing the exact same problem here - about 4 years old now, no solution:
https://www.techspot.com/community/topics/gigabyte-motherboard-stops-working.150766/
 
"I've since noticed that unplugging 8-pin CPU power connector to the motherboard, letting it stand for a few minutes and then plugging it back in and powering-on will result in the PC booting normally once"... This could be a power supply problem. Do you have another one to try
 
I replaced the PSU with an old 250W that I had lying around at home and the rig now boots every time it's powered on.

The Gigabyte PSU is about 7 months old but it's still under the manufacturer's warranty.
 
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