Don't bother with the heatsink or thermal paste. The issue you are having with the computer is related to the fan. The computer is getting too hot and without a fan to cool it down it powers off as a fail safe. If there were an issue with the heatsink or thermal paste you would have gotten an error during the hardware test indicating a bad temp sensor or processor overheating.
Also, if you do attempt to replace the thermal paste. (which is not necessary at all) make sure you clean off the old paste on the processors and the heatsink with alcohol wipes before applying the new paste.