I don't know. Maybe I see Trek differently because for me, it's a window into a hypothetical future viewed the through the eyes of several groups of people over several decades of time. I don't see it as a story about a single ship and crew, nor have I ever.
If it even remotely hung together as some sort of cohesive narrative, I'd agree.
But it doesn't. Trek is an incoherent mishmash of conflicting events and is lacking in even the most basic consistency. So continuing the 'prime' universe, no matter well intentioned, is building on a house of extremely boring cards.
What Trek got right so many years ago were the characters. The show got thousands of letters because people identified with the alienation embodied by Spock. They admired the decisive captain and the curmudgeonly doctor. Trek is Kirk, Spock and (sometimes) McCoy. That's why the new films have worked so well, because they adhere to that idea.
This doesn't mean that a Trek show about some other crew couldn't work. I'm just far less interested in it than a series about he five year mission aboard the Enterprise. I suspect that, from a commercial point of view, there isn't a big enough audience for a Trek spinoff television show. I'm not even convinced there's enough audience to sustain a show about Kirk and Spock but it seems a far better bet than the alternative.
Above all else, Trek needs to be about the people, not the setting. Nobody cares about the Federation or the geo-politics of the galaxy but they do want to see what scrapes Kirk got himself into this time.