Francis has a point. As long as the country is so deeply stratified socially, it's going to be difficult to change anything. Technically you could create better healers, but since those healers are currently all aristocracy, more or less, they would loath to do anything that removes their position. Of course there would always be some individuals who would have scientific ambitions instead of only political and financial ones, but how do you change anything if only one out of then would be on your side, nine out of ten fighting against you?
They would need to create a new middle class profession of healers, who rely on both science and magic. It might actually be easier to first create purely scientific physicians who would only attract the attention of the common people. But that'd be a pretty slow approach and even a prince would really need to work for it to stop the aristocracy from sabotaging the whole thing.
Who knows how they will do it. At least it's less hopeless because they do have a prince.