There's no particular reason the show would have to revolve around the bridge crew or the senior officers.
There was the show Space Above and Beyond where the ship's commanding officer was a occasional seen recurring character. The main characters were mostly junior officers, with a mid level officer commanding them. While we did see the bridge a few times, we never "meet" the bridge crew.
It's a similar situation on the popular show NCIS, the senior official isn't in every episode, the "crew" is a four member investigation team.
The original idea for the West Wing was that the President would be a minor character.
A new Trek show could revolve around a investigative/away team of young crew members, supported by specialists aboard the ship. They would seen interacting with the ship's higher ranking officers only rarely.