
Visible crew/equipment: After Doctor Strange experiments with the magical doorway in the Sanctum Sanctorum, he leaves it on a desert view and we cut to a shot panning across all the artifacts. At the very start of the shot on the right hand side of the screen is a crewmember standing in the shadows looking bored. Really blatant if you turn up the brightness, but easy enough to spot if you know what you're looking at. (00:56:00)
Answer: Medically, he could not heal his hands. He spent his entire fortune trying to return to being a surgeon through experimental procedures. Once he completed his training to become a sorcerer, he realised there was more to the world than he realised. The Ancient One correctly stated that he was a surgeon for his own ego; he wanted to prove he was the best and helped people for that reason only. By joining the sorcerers he was given a greater purpose and chose this over returning to his former life.
His hands were healing, just not quickly enough for his needs. This was showing his egotistical attitudes.