The United States is a prime example. The head of the U.S. Centers for Disease Control in Atlanta, Georgia is both the professional head and the administrative head of CDC, with appropriate bureaucratic supports to ensure that the administrative responsibilities are taken care of. If you look at the British model, it is different. The chief public health officer there is simply an adviser. The same is true for Australia.
Practically every jurisdiction has come up with a unique model, which is somewhat indicative of public health. We develop models that are suitable to the environment in which we find ourselves.