TRACON Departure Controller reported an aircraft deviated from their assigned heading and flew below the minimum vectoring altitude.

Narrative

Aircraft X was released for departure off of ZZZ. The standard non-RNAV departure is a 170 heading and they are asked if they can maintain their own terrain and obstruction clearance on that heading. He departed; and was climbing slow and the low altitude alert was going off while they were talking to the Tower because of the 6;300 ft. MVA and again when they contacted me. I noticed the 170 heading was a bit off but I thought it was due to the winds. I then noticed the aircraft turn due east while still in the 6;300 ft. MVA. I asked him what his heading was and advised him that he was given a 170 heading for terrain and that he was in a 6;300 ft. MVA. He told me that he had the terrain in sight; and I kept him climbing and put him on course. I did not however; say low altitude alert." My thinking he was maintaining his own terrain and obstruction clearance on what I thought was a 170 heading and then them saying they had the terrain in sight made me question if it was the appropriate phraseology at that point. In hindsight; I should've just issued the safety alert. Instead of assuming his heading was off due to winds; I should have stepped in to correct him and issued a low altitude alert."