Would a PSU still run if the motherboard is dead?

Under normal circumstances when the P.C power switch is pressed it will cause a signal to power on the Power supply. This is the same as you doing as you describe the paperclip test.

The power supply will check all voltages are o.k and send a signal back to the motherboard.

If this signal is not received the motherboard will remove the start signal and the power supply will shut down.

When you do the paper clip test you force the power supply to stay on even though there may be a voltage missing. So you have to check all voltages are there

If the voltage good signal is not there then the fans will start for about one or two seconds before the supply is shut down.

As this does not seem to be happening then the Start procedure has a problem.

If you locate where the start switch connects to the motherboard and short the two pins and the power supply still does not start then you motherboard is at fault. If it starts there is something wrong with the start switch or wiring.

It would be a good idea to check the switch on the tower first, as it is the only other component that will most likely be the problem and easier to fix than a $$$motherboard. Simply use a DMM set on continuity or a low resistance setting and don't accept if the reading doesn't change or if it has a resistance above ~5-10 ohms. Either of those conditions puts it squarely in the trash bin. If you don't have a DMM, then get out the trusty paper clip and short across the leads on the switch's terminals. When doing this, wait with the paper clip shorting the leads only long enough to verify the very first stages of start up, then remove it before the hard drive(s) get active to avoid any chance of file corruption. If it doesn't come on then and show some signs of life, then replacing the motherboard is likely your option. I hope this will help you out. Good luck!