with "GPS Primary"once Take-Off Thrust is Applied, the FMS automatically updates its position.
Incorrect. The opposite is true, without GPS Primary the Take-off update is active.
With GPS Primary there is no requirement to enter the TO SHIFT. However do take into account that if you should encounter a loss of GPS Primary before applying take-off thrust, that you will have an instant map shift. DME/DME updating will take place once airborne, but probably not in time to prevent a late first turn. So no harm in inserting the TO SHIFT anyway.
As mentioned: TO SHIFT does affect the FPLN predictions (i.e. later lift-off), so constraints could be missed as a consequence. Having said that: FMGS predictions are not to be used in the planning stage as primary means of ensuring SID gradient compliance.