UK in the 60s was doing exactly that. And many of their graduates have become high achievers in preventive medicine and public health. Some schools today will give students a fifth year when the take an MPH and graduate them with two degrees. These individuals would be prepared for leadership in public health, provided they had demonstrated leadership in other fields of medicine. Public health training does not necessarily make an individual ready for leading a health department.
Leadership comes from being responsible for a system somewhat bigger than a bread box.
Just as skill in the operating room is more than the sum of ones training, so skill in the leadership role of public health is more than what we have learned in school.