Whenever working on exposed electronics, even if everything is "known-good" and working, ALWAYS wear safety glasses and exercise appropriate electrical safety protocols.
A bad PSU is certainly possible. But, since you're not even getting to the point where it would POST if it could, there may be a problem with other part(s). At this point, we're not even looking for a display to initialize, but the ability of the PSU to power up and stay powered-up.
So (as always following power removal and ESD handling practices) remove everything but the motherboard, CPU+heatsink, the motherboard power-switch (front-panel "power" switch), and the PSU.
Connect AC power and switch on the motherboard.
If the PSU still
fails to power-up and stay powered-up, then the primary problem is isolated to the PSU, the motherboard, or the CPU+heatsink. Note: a bad CPU, bad CPU+heatsink, or bad motherboard could overload the PSU forcing a shutdown, or cause the motherboard to shutdown the PSU because the CPU is overheating*
If however, that powered-up and stayed powered-up, then the problem is most likely one of those remaining components you removed. In which case you can try testing them one at a time until/if the failure re-appears. Secondarily, it may be a bad connector/slot on the motherboard OR
a short-circuit that's somehow caused when you install a given component into the motherboard (e.g. sometimes a connector or wire can be flexed "just enough" to cause a short-circuit when an item is installed, and it's not visually obvious. Unfortunately, sometimes the failure becomes intermittent because of the changes introduced by the physical actions of installing/uninstalling parts, making it even harder to diagnose.)
To further isolate the CPU+heatsink as bad...
Removing the CPU from the motherboard may require first very carefully removing the heatsink. It depends on the accessibility of the CPU socket lever. But, ideally you can remove the entire CPU+heatsink by using the CPU socket lever. On some motherboards it's very easy while on others it's almost impossible to access the lever with the heatsink still attached to the CPU. You may wish to do that now and retest with the CPU+heatsink removed from the motherboard. If the PSU+motherboard continues to fail the CPU may
be good, but you also now know
the PSU and/or the motherboard are/is bad.
I won't go into the details and cautions if you have to deal with in-situ CPU heatsink removal (if you decide to go that route). It is risky and could result in catastrophic physical damage to the CPU and/or the motherboard. You can research that on the net.
Note: if you have or can obtain a "go/no-go" PSU tester, that can help ID a catastrophic problem with the PSU, and may
eliminate the need to remove the CPU as part of the isolation method. Simple "go/no-go" PSU testers run about US$15. If you buy one, make sure it will handle all the connectors on your PSU. Read the user reviews. Such simple testers will NOT identify all possible problems with a PSU, but if it does reveal a "bad" PSU condition (no voltage/overvoltage/undervoltage), then it's a definite fail. This can be a cheap and relatively easy way to reduce the need to "easter-egg" to locate faulty components, and of course it can be used in the future as well.
Otherwise to test a PSU you need a voltmeter, some way to put a load on the PSU, and a way to emulate the front-panel "power" switch (this is commonly called the "paper-clip" method).
I would NOT try testing a possibly bad PSU on a different, known
-good motherboard unless I considered that motherboard "expendable".
However, while it generally
won't damage a known-good modern high-quality
PSU to test a possibly bad motherboard+CPU, a short-circuit or overload caused by a bad motherboard or CPU should simply cause the PSU to safely shutdown without damage to the PSU. But here again, rather than risk a known-good PSU (and exploding motherboard parts) I prefer to use a simple PSU tester and/or a voltmeter to test (both DC and for AC ripple levels).
In usually rare cases, it's possible in a previously working CPU+heatsink assembly, that the heatsink is no longer properly thermally "connected" to the CPU. When that happens the CPU heats-up within seconds and can cause temporary or permanent failures. With some CPUs and motherboards, this kind of failure can result in permanent damage and from that point on also can cause the system to shutdown within seconds...similar to the failure you reported.
edit to add:
more info re: CPU/heatsink failures and interactions
edit to add:
more info re: nature of possible failures due to non-mobo/CPU/PSU components/intermittents