In our Earthly world, there are no new civilizations to encounter, so the field of law, however diverse, is bounded. Those who practice law operate within those existing bounded and defined frameworks to advocate for outcomes within them. New civilizations would have unknown frameworks, and indeed several episodes deal with such issues including TNG:"Justice" and Voy:"Ex Post Facto". Federation law is irrelevant to these situations; they are about diplomacy, Federation/Starfleet principles (not law), and plain ol' human ingenuity. Since these situations involve alien law, Federation lawyers would be as irrelevant as Federation law. Where legal matters calling for Federation legal expertise do arise, they concern Federation parties in more pedestrian situations, not aboard "ships of exploration". Where legal proceedings are called for aboard a starship, such as TOS:"Turnabout Intruder" or TNG: "The Drumhead", they are (1) rare and exceptional, and (2) matters covered by Starfleet regulations, with which all senior officers should be familiar. As such, there is no reason senior officers could not be reasonably called upon to advocate and adjudicate such matters. Regulations would presumably also define the processes which must be followed, so again, senior officers would presumably have suitable qualifications. Barbers, bartenders, and waiters would have steady work aboard a ship populated by a large crew; a lawyer would have nothing to do until/unless some legal matter arose that fell within Federation law but outside the scope of Starfleet regulations.
Out-of-universe: casting. You'd expect that there won't be many scenes over a season that would call for a lawyer... in TOS there were only three episodes: "The Menagerie", "Court Martial", and "Turnabout Intruder". In "The Menagerie", there was one guest star; the drama otherwise played out between the regular cast and archive scenes from the original pilot. In "Court Martial", the drama plays out almost entirely between the captain and guest cast - the remainder of the regular cast are pretty much set dressing. In "Turnabout Intruder", the drama plays out almost entirely amongst the regular cast. Unless the story centers on internal conflict between crew members (which Roddenberry did not want to see in his vision of the future), a plausible legal drama employing dedicated legal experts would pretty much play out amongst guest cast and the regulars would be relegated to set dressing, which I don't think would go over well with the actors.
