| Those numbers suggest the power supply system is OK. In fact, that the power supply probably has much more power than the computer really needed. Better would be numbers from the red, orange, and yellow wires when the system is accessing (multitasking to) everything simultaneously. IOW play complex video graphics (ie a movie) while downloading from the internet, while searching the hard drive, while playing loud music, while reading a CD ... Now confirm those voltages remain above 3.3, 5, and 12 (as you have seen). But that means you can boot the computer.|
While you have the meter, you can check the CMOS battery. Simply measure that (maybe 3.0 volt) battery. If 2.8 volts, then plan on replacing it in the next 6 months. If below 2.6 volts, replace it now. Sometimes a low battery can make power up problems. But if voltage is above 2.8 volts, battery is definitely not a problem. Better is to know what was defective before seeing the problem solved. Measure the battery first rather than just replacing it.
Power supply controller has a protective lockout function. If that function activates, then you will no longer see voltages change when the switch is pressed. That lockout function clears when removing and restoring the power cord.
Also useful would be those voltages when the system will not restart.
Normally, only a few hardware items can crash a multitasking OS. These include video controller, sound card, CPU, only some memory, a few motherboard functions and the power supply. However, only the power 'system' can keep power from also being restored. You are seeing power coming on just fine.
Obviously, when you pressed the power switch and read those voltages, the disk drive, etc all powered up. So move on to other functions.
Does the system always do nothing when the power switch is pressed? A first function executed on power up is the BIOS. With all memory modules removed, the BIOS will see no memory and beep the speaker. If it does not even do that, then the CPU and related motherboard functions are suspect. BIOS does not fail.
Typically, CPUs don't fail. But a CPU power supply (those capacitors and voltage regulators that surround the CPU) can fail. Sometimes those capacitors might be seen budging. But your sudden failure would be more typical of semiconductor failure. CPU power supply semiconductor replacement means a new motherboard.
Of course, if the CPU does beep that speaker, then the above paragraph is irrelevant.
But your numbers say the power supply system is perfectly good - no power supply swapping will be useful. Of the few items that can crash an OS, even less would cause the computer to not boot. For example, memory and disk drive are completely and obviously not on the suspect list. The CPU and its support functions (ie its power supply) are likely suspects. If the CPU does not beep the speaker, then the CPU is not even executing its very first instructions - the BIOS.