They could... but then the question becomes why? Why try to get the audience to care about new characters when there are already characters people care about in the public consciousness?
My point is that I think creativity and originality are important. Recycling characters that already exist doesn't always have desired effect- on the contrary, it can cause audiences to devalue the characters they once cared about if they're portrayed poorly.
And why continue Berman-Trek, which was simply one variant vision of Star Trek, largely created by other people? Is the point to tell an interesting story in the Star Trek setting, or is simply about maintaining the illusion -- or delusion -- of a larger continuity among all the different Star Trek series and movies -- that Star Trek is somehow real? If the point is the latter, then it quickly becomes, IMO, a geeky-ass waste of time.
It seems to me that anyone who puts Star Trek back on the small-screen would be creating a unique variant of the show. But I don't necessarily think a new series has to fall in line with Rick Berman's idea of what Trek should be. My reason for wanting to see a series that follows up on the events of TNG and DS9 has to do with me wanting to see how events I've already witnessed have impacted the fictional galaxy the people of the twenty fourth century live in.
Do I believe these events and people are real? Of course not. But I like the idea of seeing a process through to conclusion, as there are many loose threads that could be explored in the context of a new series. And as I've said before, I'd also be okay with a series set between 2265-70 if it focused on characters we've met before but don't know as much about. What else was happening in the Federation while Kirk was commanding the five-year mission?
--Sran