Rather than faulty Motherboard or Power supply, which it may still well be, another possible cause of this could be overtemperature protection. There are differing behaviours in different pieces of hardware, but sometimes when CPUs overheat, the protection on the motherboard may well just turn the whole PC off.
Try cleaning out the dust on all of your motherboard components and heatsinks with a can of compressed air preferably, or a soft brush of some kind if you have to and see if that helps. Also while you're there, have a close look at all of the components on the motherboard and see if you can find any components with obvious scorch marks around it. If you decide to take the cover off of the power supply to clean out the fan(s) and look for these burn/scorch marks, make sure to unplug from the wall and leave for a while before opening the cover.
You may also want to try running a temperature monitoring program to see what sort of temperatures your CPU reaches while gaming.