I assume this is a POWER Mac G5, which would be significant, since there was also an iMac G5... If it were an iMac G5, I'd be saying this is a bad cap issue. Still very well could be with a PowerMac G5. The symptoms would largely fit, and it's from the right general time period to have been potentially subject to the bad cap plague that descended upon the entire computing industry around that time.

ASD (and I'm not sure where you'd get a copy of that if not working for an AASP) back in those days was pretty limited in what it could do. You also didn't specify if it was the OF or OS version. Makes a difference, like the EFI and OS versions of today. And it won't replace a good old fashioned visual inspection of the MLB for any bad caps (google it if you don't know what I'm talking about).

In my experience, which I will admit has been somewhat limited in the PPC line of systems, is that when a PowerMac G5 fails, it tends to fail pretty spectacularly, and it's never worth the expense of fixing. Pretty sure that unit would be on Apple's vintage list, so unless you live in California, you're not getting it fixed unless you buy the parts off of ebay or craigslist where odds are they're going to be on their last legs, assuming the person you buy them off of doesn't destroy them with a good ESD jolt while holding it and walking across some shag carpeting or something. My guess would be you need a new MLB, and speaking from experience, that's not a fun component to replace. In particular are some screws on the CPU assy, which have these split end standoffs on the MLB. It's a big enough PITA getting the screws themselves out, but they tended to be overtightened at the factory, so a lot of times once you take them out, they're pretty much useless. They also have them sitting in some spots where it's nearly impossible to get any kind of leverage to break them free from the threads they've been sitting in, and bonding to, for years. It's a lot of work, and then once you get the whole CPU assy off, putting it back on makes it real easy to damage the MLB and the stupid little CPU connector port. IMO, not one of Apple's better engineering ideas, and admittedly one they abandoned after about a year.

If I'm right, and it's the MLB, then I'd say just scrap it. Sell it for parts, use the money to buy a new Mac Mini or something, which will run circles around that thing.