Darkness of a sci-fi drama like Star Trek should be a byproduct of theme and character exploration, rather than an end goal in and of itself. Deep Space 9 went into shades of gray territory and heavier plots because it was a setting built on various differing peoples and groups dealing with each other and what the results of such encounters would be.
Really I'd say the most interesting element for a new Trek series to build upon would not be darkness per say, but rather consequences. That was one of the main issues of Voyager and to a lesser degree Enterprise; they were shows where the characters actions didn't seem to really matter in the long term, and the effects of various episodes just weren't felt.
There is a place for serialized Star Trek show illustrating the blowback caused by the character's actions & in-actions, especially given we are in the most continuity driven era of dramatic television ever.
While I'm not against giving the characters real flaws and depth, I think the most radical thing a new Trek could do is bringing back real heroes as the main characters in an era of so many morally compromised protagonists (Breaking Bad, Mad Men, Game of Thrones, just to name a few).