Probably a more sensible approach is to say the medical field concerns itself with the operation of the human body, and how to correct malfunctioning operations.
However, public health continues to actively distinguish itself from medicine (they're different degrees and different departments in universities, for example, and you can get both an MD and a DPH), so I don't think your proposed definition for medicine will satisfy public health experts as a definition of their field. Although county health departments are often led by someone who has both a medical and public health degree, many of the people working in them will have public health training but not medical training.
Even looking at classic public health activities like sanitation/environmental health and epidemiology, sanitation and epidemiology aren't really about the functioning of the human body! They're more about conditions under which the human body is more likely to experience some particular kinds of morbidity...
Even your definition of medicine might be unsatisfactory to some people in the medical field, for example because they think mental health is medicine, or because (to pick an amusing example from my health insurer's claim form) pregnancy is not a disorder or malfunction of the body, but it requires a lot of medical expertise to manage its outcomes well. Other examples could be pain management and anaesthesia (pain, for example during surgery, is not necessarily a malfunction of the body—typically you would like the body to make you aware when it's being cut!) and elective surgeries.
Take your gun example: people should know that if they support the status quo gun laws how many people will die, or what the impact of a new lw is. That doesn’t mean the CDC regulates guns, but it does mean they should have a voice in the conversation.
A difficulty with this view, at least potentially, is that it seems to allow everything in the world (certainly almost every area of public policy) to be in scope for public health, because almost everything in the world has potential to affect morbidity and mortality. At one level this broad view seems obviously correct, but at another level it seems obviously crazy (or at least likely to bring the public health profession into conflict with everyone over weighing in on every kind of decision that anyone ever makes).
In the past there was a fight between some CDC researchers and Congress over something about handgun policy because the researchers were studying something about guns and violence that made some gun rights activists nervous that there would be an argument presented under the banner of public health to limit access to guns. I believe the conclusion was that they told CDC to stop studying this, which, again, seems to me to be susceptible of some mix of "there might be a lot of legitimate reasons to have this information" and "does the public health profession get to weigh in with claimed expertise on every issue?".
I think it's a legitimately thorny conceptual problem because if you think of health as something like "flourishing" or "people having success in living the way they want" or "eudaemonia" or something, almost every public policy question can be conceived of as a "health" question. (Or even, maybe, "minimizing all-cause mortality"?) But if someone says that the policy answer to each question is therefore obvious because a particular profession thinks it knows what would maximize things it thinks it knows how to measure, I imagine a lot of the public is going to say that this doesn't match up with its notion of what health is.
Edit: after writing this, I had this weird imagine of public health as like a superintelligent AI that's super-focused on one goal and doesn't understand, or doesn't care very much, that people might have other goals too. We think there's obviously something that's going to remain outside of the scope of the AI's interest, but since the AI is smarter than we are, it discovers a way that that thing is actually relevant to its interests according to its utility function, after all! (I realize people have also had this kind of concern about other institutions that are happy to define their scope of action more broadly over time, such as profit-maximizing corporations, or militaries. The corporation says "we need to figure out what actions will maximize profit" or the military says "we need to figure out what actions will maximize our capacity to win wars" and perhaps the scope and ambition of their actions are soon larger than anyone would have expected.) I feel like this analogy is actually weirdly apposite, at least for some audiences. :-)