Is my computer power supply broken? Details below.?

pull the big connector off the mother board and bridge the green wire to one of the blacks , with the supply plugged in should start if not its dead. 

Sounds like an issue with your PSU, I would contact your manufacturer to get that fixed if you still have the warranty for it, or buy one off of amazon and replace it. Or plug it into a different outlet or try plugging your phone charger for example into the same outlet where your computer is. (if it's working that's great, if it's not, probably an outlet issue) or if you use an extension cord check to see if it's not malfunctioning by using different devices with it and different outlets.

Could be just

 a broken power button

Could be.

One way to check it is to power up the computer with another power supply (PSU) that is proven to be working. That's what they'll do at the service center.

First it is as you have said nothing to do with RAM.

You can try what Arthur has said and that that will prove that the PSU is switching ON but just seeing if the PSU fan turns when switching on should give the same information.

Next is to prove the power switch or the Motherboard. If you can locate the 2X wires from the switch and short the 2 pins they go to that will prove the switch.

If not the switch then it is the motherboard at fault.

Are you sure that the fans do not start turning for about 2 secs. If so the PSU is the likely fault.

12 year old PC. (Year 2008). Very low performance and low value. Yes, power supply failure is likely, but could also be the system power switch. They take special supplies if you wanted to try it, but note how old the system is.

Most PSUs have a "green light" on them to indicate they are receiving power.  Does yours have one and is it lit up when plugged in?  If you're using a powerstrip have you tried plugging into a different outlet?

hopefully its not, take it to a computer store and see if they can fix it