No, public health does more to ensure the health of the public than EMS does. And that's questionable since much of public health is merely nannyism. Public health systems came into existence during the 18th century to curb contagious diseases such as Cholera, TB, Smallpox, and so on. Since most of those diseases have been eradicated (in the US), public health has morphed into a nagging nanny telling people to wear seat belts, not to smoke, not to drink sugary drinks, eat healthy, wear helmets, and vote for Democrats.
Except for Democrats, none of those are real threats to the public health.
Sanitation is part of public health, but not public safety. At least it shouldn't be.
EMS is a system designed to provide medical care in out of hospital settings and then provide transport to hospitals. We can debate if it's public safety, public health, an industry, or a hobby. Take your pick, depending on where you are, it's one or some combination of them all.
EMS is not designed, nor was it ever intended to focus on prevention, primary care, wellness, or any of the other things that people think it should be. When we try to expand into those areas, we run into other professions or industries that ARE intended to focus on those areas. If we don't think that those professions or industries are doing their jobs, then we should push them to do their jobs, not try to add to our duties to make up for their shortcomings.