It takes 4 years beyond a bachelors to earn a PH.D.
That's close to 8 years total.(It probably takes way longer when you consider you have to learn the biology of every alien species in the crew, not just human. Then what medications they can & can't take and why. So it's more like 10 to 12 yrs now)
By the time the EMH had trained anybody that far, they'd be home.
Besides beyond Janeway, B'Elanna, Kes & Seven, nobody else on the ship was smart enough to earn a PH.D any faster.