Benchtop PSU not working properly

You do realize you contradicted yourself?

Chances are real good you shorted something when cutting, grouping them... Or the power supply is sensing a no load fault, tie the +5V to ground with a 10 Ohm, large wattage resistor (or use a car light bulb) to provide a load to the +5V and see if it stays on...

 

Being an ATX power supply you need a signal on one of the wires from the motherboards connector pins to tell the supply to turn on.

What it sounds like it is doing is a self check, powering itself on when power is connected for a split second to make sure that things are working correctly, once you figure out which pin you need to ground or tie to 5 volts (depends on the type of connector it has/had) it should turn on properly.

you may have inadvertently cut the connection between these

 

I read "I also combined the leads I wanted to use" as meaning he tie-wrapped the wires he was keeping into groups, not soldered/crimped the wires together.

Also shorting the green wire and any black wire would keep the fan on until the power wire was removed/power switch set to off or the bridge between the green wire and the black wire was broken/removed. He has already experienced this "I quickly took it apart, and grounded the green wire. The PSU stayed on, and I confirmed there were the proper voltages going to each lead by using my multimeter".

I agree a short is a more probable cause than the +5v sense wire. The sense wire wouldn't usually be needed until he adds a load. Even then I have successfully run my swiftech mcp600 water cooling pump from a 4pin molex by shorting the green and black wires on the 20/24pin atx connector alone and not providing a +5v sense.
I guess it depends on the brand and design of the PSU, most have various protections, some only kick in when power is lost then immediately re-applied like flicking the on/off switch back to on too soon after switching the PSU off.

Regards,

MATT