The definition of a SID as per Doc 4444 is " A designated instrument flight rule (IFR) departure route linking the aerodrome or a specified runway of the aerodrome with a specified significant point, normally on a designated ATS route, at which the en-route phase of a flight commences. "
Note it is from the runway to a designated route.
A controller cannot authorise a deviation from a SID as any variation to the SID may compromise the design of the SID. There could be more to it than just being above an MSA such as nav aid reception or a climb gradient to reach a subsequent MSA (PBN SIDs in particular). The procedure designer has taken this into account in the SID design. Strictly speaking the controller cannot even offer "at your discretion" for a deviation since the controller does not have the authority to give the pilot the choice, the pilot is required to maintain the centreline of the procedure (SID).
As soon as the aircraft is above a recognised MSA then the controller can cancel the SID and provide a radar control service or continue under the provisos for direct routing in a procedural environment. Up till that time the pilot is on his own.