CPUs rarely fail but they can't operate if the mb is holding them in reset or failing to supply the required low voltage power rails. In my experience it's usually the motherboard that is bad and it's usually a dried out, shorted, or leaking capacitor. Most other failure modes including bad BIOS or damaged CPU tend to require help from a stupid user rather than develop on their own over time.
The power button is just a momentary contact to signal to the MB to latch the ATX PS_ON# pin low. If the motherboard detects any faults or fails to start then it may let the pin float high again and the power supply turns off. The CPU failing to boot is probably a symptom rather than the root cause. Probably.
Use the process of elimination:
Can you power up a known good MB and CPU using the scope PS? If so then PS is good.
Alternatively, can you power the scope MB using an external PS? If so then the MB is good and the PS, switch, or a peripheral is a likely culprit.
All that being said, I did some experimenting on my old MB since I've upgraded the one in my scope.
1) If the 4pin CPU 12V power connector is not plugged in then it will turn on for a second and turn back off. No beeping, even if RAM is removed.
2) With the CPU 12V connector and all RAM removed, it beeps as expected.
3) With the CPU removed it will not even attempt to power on, even for a split second.
4) Of course there is a +5VSTBY LED that is always on if the PS is on.
So double-check the auxiliary 12V supply.