Why would you pull on anything within the case other than whatever you are working on while "snapping something in"? If it is to prevent the case from moving as you press cards into slots, you should either pull on the chassis or apply counter-force on the opposite case side after lining things up. Push-pin clips are not intended to get pulled on as leverage while installing/removing other components, they are only intended to keep the heatsinks in place during normal operation.
As for re-installing the heatsink, since all CPUs in recent history have integrated memory controllers, there is no more "north bridge", only the IO hub which is without heatsink on some motherboards. For CPUs with IMC, the "chipset" does not produce much heat so I would not worry too much about re-attaching the heatsink beyond making sure it is secured properly.