Your post reminds me of a few episodes of Voyager where its strongly implied the rest of the crew are going on away missions - we just don't see it. In the episode "Good Shepherd" Janeway mentions the 3 troublesome crew members have "never been on an away mission". The plain text reading is that the rest of the crew are - we just don't see it.

As to a show structure where we want to make sense of senior crew going on away missions, but the more logical characters aren't going. I am thinking maybe a show with elements from something like 'Mass Effect'. In Mass Effect the Normandy is a small ship with a small crew, the main character is a commander. So, the main characters are fulfilling their posts as crew on the ship, but are also the away team.