PC won't power on, but can jumper 24-pin connector for power (x-post r/PCRepair)

I'm having a weird issue with my father-in-law's PC. It seemed to be the power supply because it wouldn't power on or show any signs of life.

I replaced the power supply, but I still saw the same issue. So after returning the new power supply to the store and reinstalling the old one, I realized that I can get the PC to turn on by jumpering the green pin on the 24-pin connector to ground. This powers up the PC while I have the connecting wire in.

So from there, I'm thinking it's a bad power switch. However, I tried wiring up the front-panel wires for the reset switch into the spot for the power switch, and it still would not start. I also tried again (with connecting wire) to manually jumper the power switch pins, but to no avail.

It seems that the PC will only power on when I jumper the green pin to ground on the 24-pin connector.

Does this seem like a motherboard issue at this point? How safe is it to wire up a toggle switch to hold the green pin to ground while the PC is intended to be on (likely 24/7)?