You don't need a clinical degree to work in Public health. A DrPH or PhD at some point MAYBE depending on your career path but you don't need an MD / DO etc.
I had the same question, and talked to many public health professionals at Hopkins. The reason most people had a clinical degree was because Public Health didn't really become it's own field till somewhat recently, and people in public health mostly were (still are) doctors who changed their focus from individual health to population based health ( aka public health). More and more people are now very successful in their field without a clinical degree, unless you want to also work at an individual level (eg. treat patients). But if not, you may need a PhD if you want to do research in public health or DrPH if you want to do more field work in public health in the future. I can't really think of any public health career that would require you to have a clinical degree. For status possibly, in many developing countries being a " doctor " has a different level of respect, but going with a DrPH or PhD would give you a very similar level of respect, you just won't be able to treat patients.
