Yes, it probably is why you have instant shutdowns, the CPU shuts down when it hits its thermal limit.
Is this a new issue (I would guess so)? What was done since the CPU was running without problems?
Did you apply a very thin layer of paste? Is the CPU fan actually running at a reasonable speed?
It is not a matter of applying as much pressure as you can, but the spec amount of pressure that a proper heatsink attachment will provide. With the stock Intel fans it requires pushing down the pins and turning the corner "screws."
You may need to get a different heatsink if that one is not able to attach correctly to cool the CPU. There is of course a possibility that there is a defect in the CPU that has developed.