MPH, MPA, and MPP can all be useful degrees in this realm.
Some docs I know with these types of degrees are medical directors for FQHCs doing some cool innovative stuff within their clinics related to specific patient populations, serving on local/county boards of health, have admin roles within their nonprofit hospital system or teaching roles that relate to public health and community outreach programming. We also have lots of folks in the hospital system who dabble in this type of work without these degrees, usually things like serving on hospital committees that aim to improve patient outcomes for patients with specific diseases (COPD, CHF, etc.). Certainly any physician can get engaged with advocacy at the local, state, and federal level about topics they care about, and many medical groups like the AMA and specialty organizations like AAFP, AAP, AAEM, etc. have advocacy arms where they are always happy to help docs meet with representatives.
I think primary care especially (FM, IM, peds) as well as other adjacent specialties like psych, EM, and OBGYN are the best jumping point for these types of roles (but I'm a little biased as an FM doc about to finish one of these degrees). But, there are certainly specialists who get involved with these things as well. I know a cardiologist who has been advocating hard within our health system to get better continuity of care for heart failure patients and started up a program for care coordination and wraparound services within the cardiology clinic, for example.