The PC powered off suddenly last night and since then, has shown no signs of life besides the illuminated LED in the headphones, which leads me to believe that the machine is getting some sort of power and that perhaps it is not a PSU issue. I have no other machines to test the PSU and lack the knowledge to test it alone. What is the best recourse at this point? I've opened up the case and didn't find anything off, I tightened connections just in case.
Even when the machine is "off" parts of the board are energized with +5V. (That's why you turn off the power supply itself before messing with the board.) I have seen that +5V leak onto motherboard USB ports. And I've seen the reverse--a powered hub leaked +5V back onto the motherboard--I had a power light on the motherboard with it's power supply unplugged. While I have never seen power leak in and back out I wouldn't be willing to say it's impossible.
One thing I would test: With the case open pick a small fan and carefully watch it, then push the power button. If the fan moves a bit but the machine doesn't come on the power supply isn't putting out enough power and you're getting an out-of-spec shutdown. I've seen this more than once with aging power supplies. Also, sometimes the machine will come on if you unplug as much as you possibly can from the power supply. Note that such a power supply will pass on every power supply tester I have ever seen--they only test at quite low power.