Sometimes I use children, sometimes I have a preferred villager who is the healer. But I usually don't worry about whether I have a Master Doctor or not, I find that they will often train themselves when the game is shut off or if I'm in another game slot.

I have seen villagers go to study herbs & flowers on their own, but only if they are a young teenager or if the food supply is less than 300. But even then, they sometimes get stuck by the dam.

BTW, in a new game I recently started, I didn't complete the herb puzzle. But I also have only need to cure one villager! I don't know if there is a relationship or not, but I don't really mind not having doctors if my villagers aren't getting sick! (The game has been running for a while, I have L2 explore, farming, engineering and science, completed the shell collection, two each butterflies and pebbles and six beetles still needed.)