What makes Elementary different from Sherlock? What makes one police procedural or family sitcom different from the dozens of others with the exact same format?
Let's see what makes them different - different actors, different writers, different producers, different settings, and the fact that the stories are about different
characters.
Elementary and
Sherlock are both about the characters of Sherlock Holmes and Dr. Watson, but they are differing interpretations of the characters due to differing actors, writers, and producers. The same goes for different incarnations of the Doctor. There's no reason the series we're talking about would have to have the same writer and producers as the mainstream DW; indeed, I've already expressed my opinion that it shouldn't.
I don't think the various Doctors are sufficiently different from each other to constitute different characters.
I have to wonder if you've ever seen the original series, then. The past three Doctors have all been pretty similar, but as I believe I already said, the first eight were hugely different from one another.
Maybe the War Doctor is radically different, but the rest of the incarnations are all basically the same man.
Fictionally, they're the same man. But they're realized by different actors who bring
themselves to the role. Again, the problem is that you're focused on the story and the in-universe premise of who the character is, but most people watch TV for the
actors. Plenty of people over the decades have had a favorite Doctor and haven't been interested in the others. Because ultimately they're watching for the actor more than for the character or the concept.
Even if I accepted the premise that the various incarnations are essentially different characters, I'd still have to point out that premise would have to be somewhat different, wouldn't it?
Sure, why couldn't it be? The Third Doctor's era was quite distinct from any other, because he spent most of it exiled to Earth, retooled as a Quatermass-type "scientific advisor" protagonist. Eccleston's era was also pretty distinct for similar reasons; even though the Doctor and Rose were theoretically free to go anywhere, they still spent most of their time on Earth in the present, near past, or near future.
So nothing's stopping them from adjusting the premise. Indeed, we've already got the hint of what the premise could be: stories set
during the Time War, a period we haven't seen depicted before, with the Doctor helping where he can while trying to stay out of the fighting.
Police procedurals, hospital dramas, sitcoms are all generic, but a time-traveling alien with two hearts running around having adventures in the universe while poorly piloting a dimensionally-transcendent blue box is pretty specific, don't you think?
Doctor Who has been around for 50 years, had over 35 seasons on the air, and has had eleven lead actors, multiple producers, and varying formats. I'd say it practically constitutes a subgenre unto itself.
(Not to mention that many of the details you mention are not universal across DW. He wasn't clearly established as an alien until his second incarnation, wasn't revealed to have two hearts until his third, was confined to Earth for most of his third, and his ability to pilot the TARDIS progressively improved to the point that his sixth through eighth incarnations could almost always navigate perfectly, though the modern series has gone back to giving him the occasional navigational error.)
Wouldn't something new, something different, have to be added to the premise to make it distinguishable from the original Doctor Who?
I'm sorry, I have to laugh at that. The modern series is already hugely different from the
original incarnation. In the original
Doctor Who, the primary heroes were a pair of human schoolteachers lost in space and time with a teenager from the future and her eccentric grandfather who had great wisdom but tended to get them into trouble, and they had adventures designed to teach children about science and history. Then it became a series about a clownish cosmic tramp wandering the universe with good-looking young companions, running from all sorts of dangers
Scooby-Doo-style but then outsmarting them and taking advantage of how much they underestimated him. Then it became a series about a swashbuckling scientific advisor working with the military to protect present-day Earth from alien invasions. Then it became a succession of Gothic horror stories whose protagonist was a Bohemian eccentric, and then it became a more farcical comedy-adventure with a broader version of the same lead, and then it became an ensemble drama where a more insecure and vulnerable hero traveled with a mismatched family of young people who kept getting him into trouble in one way or another... and so on.
So the idea that
Doctor Who has ever had a single, unvarying identity is completely ludicrous. Like the Doctor himself, it would never have survived as long as it has without the ability to reinvent itself.
That's not been my impression. My impression is that fans want a whole new series (not specials, not one-offs, not webcast mini-episodes) starring McGann right now, while Capaldi is playing the Doctor, too.
Go back through this thread, please, and quote me a post where someone expresses that specific desire. The cool thing about BBS threads is that you don't have to rely on "impression" and memory -- you can click back through the pages and reread what people actually wrote.
I've seen every incarnation of the Doctor in action, and I'm well aware of the differences between them, but I'm more aware of the overwhelming similarities. 
But I guess only superfans count?
