I would have guessed an obscure little institution known as the Roman Catholic Church just might’ve kept the pronunciations alive, maybe...
Actually, there are significant differences in the pronunciation of liturgical Latin, which the Catholic Church uses, and classical Latin, which is usually taught in schools. When President John F. Kennedy said, "civis Romaus sum" (I'm a Roman), he was using classical Latin. When you hear Mozart's "Laudate Dominum" (Praise the Lord), the liturgical pronunciation is used.
Up until the eighteenth century, classes in most European universities were conducted in Latin--and students also had to be fluent in Greek. For oral exams, a student might have to respond using Greek to a question asked by a professor using Latin, or vice versa.