If the CPU is overheating, the machine would likely not just "black out".
It would likely BSOD and reboot. (I doubt less tech savvy users would disable the Automatic Restart.)
Most likely not the PS.
Does the system boot/run and display fine in Safe Mode?
Boot into Safe Mode... and let it stay in Safe Mode for an extended period.
If all is fine, it's most likely a config issue.
When the screen goes black, what happens if you tap the power button (not holding it in for four seconds).
If the machine gracefully shuts down, that tells you it was still within Windows... and had not crashed.
The display (alone) had blacked out.
That would put the issue on the onboard video or config/drivers.
If onboard video is the only defective culprit, the dedicated video card should resolve the problem.