I think if you're telling a Star Trek story at some point, you need your Starfleet Captain and Starfleet crew characters. It might be interesting to see a series that changes location a lot, so that in one sense it feels episodic (planet of the week being planet of the year) but I don't see a need to remove the anchor of the Trek captain and crew. If variants of those characters are going to appear every year, why not just make them the same characters?
In Trek literature, we've seen a number of variations on the familiar "captain and crew" formula. SCE/Corps of Engineers
focused on an engineering team as the main characters, with the captain and bridge crew of their ship as secondary characters. Vanguard
's main characters included some familiar roles like a starbase commander, a science officer, and a doctor, but also a JAG officer, an ambassador, an intelligence officer, a reporter, a civilian trader, a Klingon spy, and an Orion merchant prince. Then there's my own Department of Temporal Investigations
, which is along the lines of an FBI procedural whose main characters are DTI employees including field agents, administrators, a temporal physicist, and a counselor for the time-displaced. And of course there's Keith R.A. DeCandido's Articles of the Federation
, a West Wing
-style novel about the president of the UFP and her staff, characters who have continued to appear in multiple subsequent novels.
Over in the comics, there have been other well-received departures from the ship-and-crew formula, particularly John Byrne's comics for IDW, which include an Assignment: Earth
miniseries (in the vein of what a Gary Seven TV show might've been had it actually been made in the '60s/'70s), a saga told from the perspective of the Romulans, and Leonard McCoy: Frontier Doctor
, detailing Bones's adventures as a civilian between TOS and TMP.
So there's ample evidence that all Trek series don't have to follow the same formula of being centered on a ship and its command crew. It's been done numerous times in tie-in literature, so there's no reason it couldn't be done on TV.