Yes, the surgeon has the right job, in fact I have 2 surgeons, I switched them but nothing happens. I didn't really do anything to make it work, I kept spamming the Wait button (24 hours, sometimes 26) and it just happened to work all of the sudden. I kept buying the supplies needed while they were being healed but then it stopped again.
The weird thing is that most of the time the game doesn't even ask for the money that the crew costs. Then it started asking for the money again, but no healing.
I really hope it's just me being dumb and not a bug, because I was enjoying the game so much...