Hey i have that same supply.
Very long cables, but not so beefy in specs as a 360w should be.

Did you load it up enough?
Also are there no tiny solder cracks, like a curly ring inside the solder?, also no leaky or bulged capacitors dc filter.
If you blow it with a heardryer straight on the components for some minutes until the whole thing is really hot, does it still do the same?

Maybe the filter caps are bad, and it detects and overvoltage on the feedback to the regulator, or some of those power resistors are bad, and it automatically shuts down to prevent damage.

If you are talking about PS-5551-1F (560W)

Everything seems perfectly OK.
The only noticeable thing is black dust from components that heat up when in use.
I hear click sound when i flip the switch, another click when the PSU starts from the button, than another click after 2 seconds the PSU shuts down.

When i jump it alone with only multimeter the PSU works with out stopping. Voltages are all OK.
As soon as i add devices, such as CD ROM and HDD to it to test it it shuts down.

I would blow away the dust with compressed air and afterwards clean it up with alcohol.
Then i would resolder all the suspect bad solder pads, especially the ones with much weight on them like the fet's and the diode's attached to the heatsinks, also would replace the filter capacitors on the 3.3v, 5v and the 12v lines with brand low esr ones.
After this you most likely would have a working psu.

Also might be a case that there is VR pot meters for adjusting the output voltages and they corroded a bit and changed in resistance, this could cause an undervoltage at the output at load, maybe this causes a shutdown.
I would read their resistance, drench them in alcohol, wiggle their settings and reset to their original position with resistance the same as before.
then ad a fdd or cdrom and check the 5v and 12v if they drop under 4,5v or 11v i would adjust their settings and add afterwards two extra cdrom drives, and recheck.
Sometimes several psu do start with 11.5 and with more load they up to 12.2 or more, so be carefull and test while adding load.
Also some psu's tend to overvoltage at almost no load, so attaching only a hdd can cause its death.
CDrom drives several 4 or such would be enough for such load test and they do not die easily, hdd's do when the 12v is too high the servo motor ic can burnout.

As for dust and bad solder, its dust free and no solder joint seems to show any kind of deficiency.
I will check for pot meters.

Also i tried it with 4 CD ROMs and an HDD when i wrote it shuts down. Seems that when a high load is on the PSU it shuts down.

check the 12v rail voltage when you load it too much.
Also pic of the top of the pcb please, especially the part with the 5 or 6 capacitors near all the power cable.

I helped my co-worker with an psu atx200w it had same symptoms.
When run with certain load the thing shutted down.
We tested the esr of everal teapo capacitors and they were in the 10 to 15 ohms range very bad for 2200uf caps.

Try with blowdryer and heatup and see if it lasts a bit longer then change the caps.

Well, thing is, there is no cold solder 😒
Every thing IS perfectly soldered.

https://www.dropbox.com/s/y9a6sshhrll38mc/SDC12499.JPG
https://www.dropbox.com/s/h1ar6wi8v9n6cqv/SDC12500.JPG
https://www.dropbox.com/s/1ihisq3sfgf058q/SDC12501.JPG
https://www.dropbox.com/s/ko9n4wbv50zmh8o/SDC12507.JPG

The thing is not really about cold solder but failing filtering capacitors.
They seem to look rather ok, some plastic is shrunk a lil by heat.so they already endured some power hours.
The can have internal increased esr value and causes to behave badly at filtering the pwm dc signals on the 5v, 12 and or 3.3v lines or only one could be bad.

When there is enough load supplied the cannot handle the ripple that would occur and the power gulator would shut down as fail safe.
Heating the capcitors hot can cause them to lower the esr value a little to cause thm behave a little better.
See if you can add one or two extra devices, and it survives.