Pookie, just about every one of my villagers become an esteem elder (or the equivalent after the first 50), including the doctors. In fact, I have several villagers who become master doctors late in life after they've already mastered 3 other skills.

Once one of my villagers masters 3 skills, I change their clothes to a particular outfit (just for fun) and switch them to doctor. I make sure they are trained to be at least a doctor trainee.

I've found a number of these villagers actually do cure sick people, especially when I'm not playing. I've also found when I am playing, frequently it's the older master doctors that come running to cure someone who's sick. Their only drawback is when they're elderly, they run so slow that someone else has already cured the villager before they get there.

I actually start much earlier with my doctors. Each and every villager is trained to be a doctor trainee as their first skill. If possible, I try to use the kids to heal the sick people. That gives them a head start on being a doctor trainee once they're 14.

Unless the kid has another skill inherited from his/her parents, I select the doctor part in the details page. That way, if they turn 14 while I'm not there to train them, they will begin to train themselves. Then, when I get back to them, I switch them to one of the other skills so they can master on those. Once they master 3 of them OR become elderly, whichever comes first, I switch them back to being doctors. That's when some will come running and master the doctor skill while others seem to have not bothered healing anyone (most seem to have healed at least a few villagers).

The times in which I tried to get a young villager to become a master doctor, I get the same response you get. They do great studying the flowers until they gain Adept status. At that point, they really won't do anything other than play in the pond and goof off. I don't try to stop them from goofing off. If a villager does not make any progress for awhile, I switch them to another skill. Then they can go back to being doctors after they've mastered the other skills. When they're older, they much more quick to heal people which is why they can master out as a doctor even if they started that skill very late.

I don't try to force them around to research the flowers because I just don't have the patience to do it. Plus, since every villager is trained as a doctor trainee from the start, I don't really have a huge need for any one villager to become an actual doctor. With 115 villagers, I always have a good 20 villagers who have already mastered their 3 skills and are turned into doctors so anytime an event occurs that make the villagers sick, I have doctors tripping over themselves trying to heal them. Sometimes it's a race to see which doctor will get to the sick villager the fastest (sometimes they're so fast, they beat me when I'm trying to find a kid to heal the sick villager).

Oh, and I never use sick stews to force anyone to become sick so a doctor can heal them. It just works as the game plays out.