Except they're not independant of one another. I'm sure the behind the scenes show DW: Confidnetial mentioned several times that the new show was considered a continuation. Previous incarnations of the Doctor from the 1963-1989, 95 era have been mentioned or seen. i.e. The Fifth Doctor in the minisode "Time Crash", and what about "The Day of the Doctor" where we see the current Doctor, eleven previous incarnations and one future. I don't get how they can spell it out any more clearly that this show continues on from the 1963-1989, 95 era.
New Star Trek
isn't independent of old ST either. DeForest Kelley cameoed as McCoy in the TNG pilot, and its second episode was a sequel to a TOS episode. The modern Trek shows and movies brought back every lead character from the original series except Uhura, as well as Sarek, Kor, Kang, Koloth, Arne Darvin, Zefrem Cochrane (recast), etc. The new shows focused on new crews and ships (or stations), but they were direct continuations of the same universe and continuity.
I don't know why you're treating this as either/or. A sequel is, by definition, a continuation of its original. If it weren't meant to be a followup on the same reality and events, we wouldn't call
it a sequel. If a show revives an old concept but isn't
a continuation of the same reality, then we call it a reboot, like Battlestar Galactica
Well, for the most part. There are a few out-of-continuity sequels, like Halloween III
. But as a rule, sequels are continuations and continuations are sequels. So there's really nothing to argue over.