They sell PSU testers for $15-20. I bought one years ago and it still serves me well. You plug in the 24-pin and the other power connectors and it will tell you where it’s faults lie. If none, then the motherboard should be scrutinized next.
Capacitors exist all over your system. They are on nearly every component. Look for them and then look for the bulge out the top. There are reliefs (a big X) in the top of the capacitor to allow it to “blow” out that direction. Sometimes it only bulges and does not blow, but that’s enough for failure. If you open your PSU, try not to root around too much with your fingers. Be cautious of the power load that might remain. Visually inspect the caps. It might only be a slight bulge or none at all, but the caps can still die.
I’d lend you a PSU, so maybe you have a friend close who might too? You may still consider buying a new PSU to allow for future upgrades and put the old one on the shelf as a spare or for other purposes. That is, if the PSU is fine. It is still possible the motherboard is at fault, but again with these symptoms, it would likely be a cap.
Good luck!
I’ve had PSUs with odd issues that still worked for the most part, but when I tested them (with my tester) they reflected a weak voltage on one segment or some other failure. But it would still sorta work. These testers do work.
Mine is very similar to the one in this Corsair PSU testing guide. I think it was $15.
https://www.corsair.com/us/en/explorer/diy-builder/power-supply-units/how-to-test-a-psu-power-supply-unit/