Medical Director starts with a weapon permit.

[Frank_Stein]

The Captain and Detective are both security related, while the MD is not. So, canon wise it would make sense for a civilian role like the MD or the Bartender to not have the right to wield a weapon. I figured both were examples of them smuggling something onboard previous to the round start.

Gameplaywise you have a more solid argument, but really there's no reason either should have to start with a gun when so many other roles don't, so I would just consider the fact that you get something a nice little bonus and leave it at that.