|Although the psu specs seems OK...|
Output +3.3V@30A,+5V@36A,+12V1@20A,+12V2@18A,-12V@0.8A, +5VSB@2.5A
I suspect it's a cheapo power supply. A good un for comparison is
All the ON/OFF button does is momentarily connect pins 6&8 on the motherboard (Intel Standard); an ohm meter will test it. It's normally open and only closed when button is depressed.
Likely is saying that fans spin, lights light, but nothing else happens. Typical conduct of a pooched power supply.