Thank you all for help and suggestions. Before anything else I'll have to clean the desk, and cobble an isolation transformer from 2 back to back normal transformers if I want to use the oscilloscope. Cleaning that workbench might take days!
Meanwhile I've searched more for a schematic. Could only find other ATX schematics from older models and max 200-300W only. The surprise was that they all have a post regulation
mag-amp stage for the 3.3V (3.3V is one of the defective rails). I've read that more recent ATX models might use transistors instead of a mag-amp.
On the outside sticker it says max 24A on 3.3V, max 24A on 5V, max 66A on +12V, max 05A for -12V and max 3A for 5Vstb. Not sure yet what kind of 3.3V stabilization uses mine, but I can not identify which one would be the magamp coil, maybe mine is using transistors instead of mag-amp.
The idea with the magnetic amplifier is that a saturated core shows a much lower permeability, thus a much lower inductance. For high frequency an AC this will look like a closed switch. When the core is not saturated, there is a big permeability, so big L, and a big L will stop AC current. We can saturate the core using a small DC current (passed through many turns) in order to stop tens of amps through another coil with less turns. Thus the magnetic core can be used as a switch, as an amplifier, even as a memory.
Found a seminar from former Unitrode (now TI) that explains the mag-amp very nicely in the first 2 pages:
Magnetic Amplifier Control for Simple, Low Cost, Secondary Regulation by Bob Mammao
https://www.ti.com/lit/ml/slup129/slup129.pdfWhile this idea of taking advantage of the magnetic hysteresis looks very interesting, the history of the magamp is even spicier, going back to WWII, V2 rockets and the operation Paperclip:
The Vacuum Tube’s Forgotten Rivalhttps://www.nb9m.com/index.cfm?key=view_news&TransKey=D5623FA0-C14B-4F99-AD74-0E38C60CEDE2&Small=1 (the page is missformated in my browser, the rows show truncated, I had to CTRL+A to select all then copy paste into a text editor to read it in full)
The magamp idea seems very clever, and I didn't know about it before, so now I'm tempted more to experiment with magamps than to try fixing the power supply.

So far only -12V and 3.3V are not working. All other voltages are OK even when a 45W/12V car light bulb is attached on each of the +12 and +5 rails. It start for 2 seconds then shuts off. To answer a few direct questions:
Question here would be, what load does need to have a 800W Power Supply?
I guess half of it would be enough. The desktop is an i7-4790K with 32GB RAM, a single nVidia GTX760 and 1-3 HDDs. No overclock, no gaming. The 800W choice was to make it run colder, thus less noise and less dried capacitors over years. Can also power the PC without a hitch over mains power surges of about 1-2 seconds.
Does it not start at all, or starts sometimes? For example, after staying disconnected from mains for, say, a day or more, or after the input capacitors are discharged?
I don't suspect the capacitors. It died at a wake from standby, the speakers made a pop lawder than usual. Now it starts for about 1-2 seconds, power good comes up for a fraction of a second, than it realize it's missing the +3.3V and the -12V, and it shuts itself off.
symptom was, it would click on and immediately turn off. Interpretation: primary-side power stages are all working, at least somewhat. Probed around some things, found 3.3V output was overshooting, hmm. With -- I forget if I checked it under load, or that was sufficient evidence?, I identified that was the problem; the power monitor chip (which fortunately has a datasheet) was faulting out. Okay, trace 3.3V power path. The magamp regulator was saturated full on. Well that's no good. The BJT driving it had failed somehow. No visible damage. Replaced with an in-stock part, good as new.
Very similar symptoms, thought mine might not have a magamp, but some MOSFETs instead, not sure yet. It's very hard to even see if on the radiators are MOSFETs or double diodes. They are covered and only 2mm or so space above the markings.
Measure the resistance between each rail and ground.
It's increasing from 0 to about 3k in 10-30 seconds, except the 3.3V which is increasing much slower - after about 2 minutes was only half of those 3k and raising. Looks like charging capacitors up to the max voltage supplied by the ohmmeter.
What am I suppose to look for while measuring with an ohmmeter?
Is it just me or the display of my notebook I am currently using, or are there several areas of the PCB that look like they have gotten too hot and/or dirty?
The pics were with a flash will try better ones in daylight, but it was indeed something splashed there, I suspect those are some flux residues from the heavy soldering. It even has a metal plate solder on top of the PCB traces.