It may be failing power supply.
Your probably right on that.
But Im still confused as to whether its a case of the processor fan failing to keep the processor cool so the PC shuts down, or the power supply fails.
The discription is not clear enough to solve this problem effectively.
Altough resetting the power supply by removing the power plug, could mean a failing voltage regulator, or an over heated one.
As for a fan that stops abruptely, try cleaning the area around all fans with compressed air, making sure the fans do not spin during this process, as this can cause the fan to create a voltage going back into the motherboards power circuits.
Be carefull when cleaning power supply fans, as the componants inside can hold a lethal voltage, even when the power plug is not plugged in.
Blowing dust out of the power supply unit will help a lot as well.
Good luck to you.