Especialy at the beginning of the game, you can't let your villagers alone too much!

The doctors will never heal anybody by their own if they are not checked as Doctors. That's like that since the 1st VV!
And it's right for the others skills, more or less. When they are Master on something they tend toward of doing it by their own.
I never quit my game without checking at least 2 doctors and 1 farmer! (it may be more, depending of how many villagers I get)
Sometime, you may see a villager who LOVES doing a task. Even if you check and put him on another task, he'll go to do the one he prefers, time to time!

I think even the LDW team doesn't know why and is still amazing about that!

Me, I love it!