@dlloyd I tried the same wiring I had before but with 2 diodes from the +5V AUX and +5V to the PSON but the power supply still starts only a second. 
For now, I will probably continue to use it with the motherboard (even if it takes much place on my workbench) but I will also search for a standard ATX power supply to reuse from an old computer, I think it should be easier this way.