so I'm not exaggerating. Actually, I guess you could split one of the burrow bits into two scenes in half-blood prince, so perhaps that one has 3. A dozen settings is a very, very low number for a 6 book (so far) series. That's not good or bad, it just shows a reluctance or lack of need for more locales.
But by your argument, then she's not using those already established scenes either. Here's all the ones I can remember.
Diagon Alley: at most, 1 scene per book (not in every book)
Hogsmead: at most, 2 scenes per book, usually 1, and not in every book) (and this is pretty much an extention of Hogwarts)
The horrid aunt and uncle: 2 scenes in one book, usually only 1.
The cave: 1 scene in one book. This one has the highest potential to be used again as a location for a horcrux search.
The burrow, 3 scenes in one book, one or two (usually very brief) scenes in most books, no scenes in book 1.
The quidditch championship field, one extended scene in the whole series.
Ministry: one scene at most, usually none (though it does have one quite extended scene)
The bank: one scene at most, usually none
Train station: once per book. Possibly twice in one, but I can't remember it.
The zoo: one short scene in one book.
The park: one scene in one book (when the dementors come for him).
The Riddle home: two scenes in one book, and then only in a memory.
potential for any of these besides that cave as a scene where searching for the horcrux: very very little. Except the bank. That'd be amusing to have to break into there to get at something. The ministry would also be interesting, but it'd be a jump to convince me Voldemort reasonably stored a horcrux there.
I'm sure there'll be one or two scenes at the Burrow. Perhaps they think about plans there. There'll be 3 or 4 scenes in other place to either retrieve a horcrux or find tehy were wrong. But the guy with the necklace destroyed that horcrux, presumably without Voldemort's knowledge. And the diary and ring are destroyed. That leaves us with 4. If one is harry, that means three to find. Reasonably speaking, I'd say at least one, probably two of those are already at Hogwarts somewhere (there's problems with that theory, but there's problems with lots of things). That means a lot of time at Hogwarts, doing research and exploring.
So, no, I don't think he's going to spend a lot of time away. There's no better place to do research than Hogwarts, and I think at least one other horcrux is there.
My theory is that one of them is in Azkaban. It's been talked about so much, time for us to see it.
I'm not sure I accept the premise that Harry is a horcrux. Tell me when/how Harry was made a horcrux? It wasn't before Voldemort cursed him, or else he wouldn't have tried to kill him. It couldn't be after, I don't think, since Voldemort disencorporated at that point.. when else did he do it? After he was brought back? Most of what I can think of doesn't seem to make much sense.
edit: oh, there's the lighthouse, one scene in one book, and the house of that old lady, one scene in one book, in a memory, and the train: usually one, sometimes two, scenes per book, always pretty short. I guess you could argue that a few of the shops are different locales than Diagon Alley, but then you have to break up Hogwarts into a dozen million different settings, and the porportion of scenes gets weighted even more in that direction.