On my Air the fans were running a lot, but no overheat shutdowns.
I replaced only the thermal paste and after that that the fans rarely ever turned on.
Trouble putting the cable back in the ZIF socket?
Get some electrical tape and cut it down the the width of the ribbon cable fold over one end to create a pull tab. Now place the tape onto the ribbon to give yourself something to pull either in or out as needed. Then you can peel off the tape.
Don't forget the unlatch the connector first!
Source: Re-connecting fan ribbon cable