If I recall, ST used three shifts, with most of the senior officers on the “day” shift, and junior officers taking the less active shifts. The Captain was always on the bridge during that “day” shift, but delegated command to his commanders during the other shifts. It seems like they tried to plan their crucial missions into that “day” shift to use their most experienced crew, or summoned the senior officers to relieve the junior ones in times of crisis.
Considering the small size of our ships compared to those of ST and that we (almost) always dock at a station at the end of our shifts, it could be that we really only have one active crew that goes on missions. This would also explain why we never hear about any other shift’s missions in our briefings.