There would be a host of folks who stood watch on the bridge, not the same bunch over and over. They must be exhausted doing 24 hours on, zero hours off watch rotation.
In the Coast Guard we call them Boarding Teams (there is no partying, goldarnit), and they are headed up by Boarding Officers; I was a team member as a young seaman (and later as a petty officer on another ship); the boarding officers were young junior officers or the occasional chief. Thank god our Captain, XO, or Ops Officer never went on boardings with us!
That’s what is funny about the series…helsman on a ship is less demanding than on a star ship, so it wouldn’t make sense to have a third class quartermaster standing watch up there.
Also, on my aircraft (the C-130 Hercules) there are between 5 and 8 enlisted personnel, depending on mission, and only 2 officers (pilot and copilot); they were handy to have on board if we were diverted to some other airport during a mission and we had maintenance issues, as they were going to fix us.
Well, when they were, it sure as hell wasnt on a Ship of the Line, like the Enterprise. At most a Frigate (cruiser) but more likely just a Brig. Cook was only a Lieutenant when he sailed on the Endeavour, later as Commander he had the Sloop HMS Resolution , then promoted to Post Captain. Bligh similarly. Of course, they’d be called “Captain”.
To be fair, the Enterprise also had Diplomatic, “Showing the flag” and even, yes, Military missions also. In sail days those would be handled by a Frigate, in oil and coal days by a cruiser.
Indeed the Enterprise is considered a “heavy cruiser”, not the equivalent of a ship of the line or a battleship. The Federation built a single prototype Dreadnought but ditched the idea of having a purely war-fighting ship.