I assumed that they had one surgeon for each shift plus specialists so Helen Noel as a psychiatrist may or may not supervise sick bay in a crunch depending on her general medical experience. They probably have a pathologist too and it's debatable to what extent there is integration with life sciences so you might have bio-physicists, biologists, exo/astro-biologists, virologists, and geneticists etc on the research side of the medical team.
It's also worth noting that if Chapel became a fully qualified MD by the time of TMP then should would actually have been a trainee doctor rather than a nurse for pretty much the whole time she was in TOS.
And finally, the doctors are also officers, and would no doubt be given tasks to do as officers that have no bearing on their medical training. Being on the bridge as a general advisor would probably fall into this category.