I'm not entirely sure but I think I remember hearing that it is a civilian music instructor who leads the bands. As for the drill and flags I would assume it would be a NCO with great military bearing and experience. Part of the military is looking and sounding good while doing something. Having someone who is not musically trained, or proficient at teaching, does not help the Navy achieve this goal at parades and PIR. I, however, have no factual evidence to back-up these conclusions. When my dad was in the Air Force when dinosaurs roamed the earth
he had musician instruct them for band.