Dear Jeff,
Only briefly comment here because, as a emergency medicine doctor, I could really go down the rabbit hole here on RPG modelling of Medicine as a career, or of the pathophysiology of injury and illness. (Closest I've seen is the work done in GURPS 4th ed BioTech.)
A few points ...
For role-playing purposes, Paramedicine makes much more sense as a career than Medicine itself. I can see ambulance officers, nurses, army medics, etc, getting into typical PC mischief. Properly qualified medical professionals? Not so much. (We tend to do quite well financially having nice, steady jobs and avoiding dangerous, erratic situations.) A doctor serving a crew of 5-6? A waste. More realistic would be a nurse for passenger/crew manifest of 100 or more (and just someone with some certification if less than that), and a doctor once you get up over 1000. Even in the armed forces, the doctors are more likely to be found in field hospitals well behind the front lines. As for "(medical) Researcher", just treat that as a subspecialty of "Scientist", rather than of Medicine. Full-time medically-trained researchers are quite rare, and are a very different beast to clinically-practising doctors. Though, of course, many practising doctors also do some research on the side (in which case, treat that as a second career or dual-class).
To keep it simple, the core skill of "Medical" would actually be "Diagnosis", which there would be various optional specialisations of. Treatment with medicine is pretty much a no-brainer once the correct diagnosis has been made, so for most medically-managed problems, the diagnosis and treatment roll are one and the same. However, some conditions would need to be treated with "Procedures", with either mandatory specialisations, or separate skills: Surgery, Psychotherapy, Critical Care, Imaging (taking the pictures, not their interpretation, which is part of Diagnosis), etc. Unless one has trained in that specific procedure, you just cannot offer it (or else will be practising it at a default-level skills ...). Some of those Procedures may be offered by stand-alone paramedical professions who have only a passing familiarity with Diagnosis. A "good doctor" would actually have very high levels of interpersonal skills (esp. Diplomacy and Teaching, though Fast-Talk is occasionally needed in emergency medicine!), rather than necessarily stellar levels of "Diagnosis" or "Procedures". I have no idea who the "best surgeon" in my hospital is, though I can tell you who is the most thoughtful and considerate, and seems to be be most interested in his/her job!
As for (Dis-)Advantages, the only near-obligatory one would be some level of Wealth. Mingling among a social circle that is mostly doctors, I can tell you that almost every A/D is on display there somewhere, but none of them are near universal (except 1-2 levels of Wealth), though Workaholic or Code of Honour might be close. The suggestions in GURPS: Traveller are a reasonable start, though playing more to a stereotype than to reality (Disease Resistant, Pacificism, ...).