Diagnosing a no POST Case: PS, MOBO, CPU, other?

Looking for advice on how to proceed with diagnostics. Win10 dual boot with Win7 for occasional use.

Symptoms & diagnostics
In approx last month, on occasion unit would not wake from sleep prompted by mouse input: system would crash as if power was turned off.
At occasional start up system would halt, again as if power had been cut out; after another push on the start button, the system would boot to normal.
Had one or two instances of running unit just powered off out of the blue, as if power had been cut.
Failure event: start up press of start button did not cause start up, pressed 2nd time and had the impression of a slight peculiar sound like a short, but could have been the start switch - regardless unit dead.
Opened case to use on-MOBO On switch provided on this MOBO; noticed light under this switch (orange glow) was off. MOBO switch ineffective. Conclusion: suspect PS, MOBO short, or cable problem.
(Yes, I know operating with a flaky PS is not a good operating practice as this has the potential to spike components, but I was under a pressure with a time critical task.)

Pulled PS, did live test with 2 MOBO power cables connected and PS start pins (24P P16-COM) jumpered: PS fan runs and Voltage output measures good on all pins.
Question: Wondering if PS needs to have outputs loaded to verify PS is actually good?
Rationale: I noted that on-MOBO ON Switch remained dark after reinstalling/cabling up PS, unit still dead. However, after pulling all extraneous hardware connections (loads) to MOBO (HD, SSD, Graphics card ... ) I now note the on-MOBO On switch glows as is normally the case when the PS is switched On. Can't hear PS fan running, and on-MOBO On is switch ineffective (PC won't even POST), so have to wonder how some power is getting to the MOBO. Maybe fault (short) protection of PS allows some low current supply (3.3V) to the the MOBO but disables other outputs (12V).

Thinking next steps are to pull CPU Cooler, then pull MOBO and do a thorough close up inspection. At this point, no obvious problems on MOBO front side. Then pull CPU to inspect (thermal paste contamination on pins?) and reseat. Also inspect traces as I've always been careful, but nervous about the heavy cooler hanging cantilevered off the MOBO.

Very, very helpful comments Ghot. You've pulled me back from rummaging further without addressing the obvious 1st diagnostic priority; confirm good power. Even if the local BBY can't test tomorrow, they do have an inexpensive PS in-stock (immediate temporary solution, and handy to have on hand). Thank you for your rapid response: I'll sleep better tonight.

In 2013 I built a computer for my wife with a GA- Z87-D3HP and a i5 4670K, a computer almost like yours.
In 2018 the computer began to shut down by it's own. It started to happen once a week and then began to get worst till it ended not booting.
When I took out the MB it has a little smell of burned paper.
I couldn't find where the smell was coming from, but I suspect it was from the CPU mounting socket.
Microcenter asked US$ 50 only to evaluate the MB so I decided to buy a used Asus Z97 MB on eBay for US$100 and the computer came back to life and it's working very well.

Failure was in the MOBO VRMs. Haven't pulled the heatsinks, but I found signs of leaking capacitors.
Based on some research found the lighting up of the start button probably has to do with PS supplying a standby +5V despite the PS not being "on."