My population is at 101 and I have had no trouble training doctors. I literally have dozens and dozens of Master Doctors. Of course once a villager masters in one area I immediately switch their preferences to another field so that everyone has a well-rounded education. There are many ways to play the game OptiTron. Sounds like you chose to educate your villagers a different way. Here's a tip:
Thank you redneeckgirl for the tip about the corner of the hospital, I don't think I was aware of that area.