Could be either mobo or PSU, really.
I would, in this order:
1) Replace CMOS battery (while unplugged), then do a CMOS_CLR, then plug in, and attempt to power-on.
2) Try swapping a known-good PSU into the system.
3) Try swapping mobos.
But yeah, that's a wierd problem, but I have a hunch that the problem has something to do with the +5VSB of the PSU, and the mobo's power-on circuits that depend on that. CMOS battery could also be involved.
Also, check how many USB devices you have plugged in, and if possible, unplug them all except for keyboard + mouse.
Also, if using DisplayPort, change DP cable for one that is VESA-compliant (doesn't pass power on pin 19). (Bad / non-compliant DP cables can affect +5V line in PC, which may be tied to +5VSB line if USB ports are set to provide power to charge devices while PC's power is in ATX soft-off state.)
I've had a couple of buddy's rigs, that stopped immediately powering-on, there was a "delay" of seconds to minutes powering on from pressing the front-panel power switch. Sometimes, unplugging or switching off the PSU in the back and waiting and switching it back on would allow the PC to power-on. In both of those cases, replacing the PSU (*weak / bad +5VSB from PSU) fixed the issue.
But you mentioned trying the unplugging trick.