Being the Avatar ties you to all of the elements. Korra is the Avatar; I imagine her offspring could potentially be Benders of any kind, even if she was originally from the Water tribe.
I don't think it's supposed to work that way. Avatars have access to all the elements, but each one is part of his or her nation, which is part of the point of the cycle. If every Avatar were "element-neutral," that would kind of eliminate the need for the cycle where each successive Avatar is part of a different nation, each one bringing their people's own distinct elemental character to the cycle and being different from the Avatars that came before.
Perhaps, but as the Avatar universe starts to blend together, how will that affect the Avatar itself? For example, in TLoK, we've already seen Bolin and Mako, brothers who were born with different bending abilities, as well as Tenzin and his sister. What happens if the next Avatar is born right in the middle of Republic City? The world is changing, and the different nations are not as separate as they once were. What happens to the Avatar spirit as the "pure" bending lines become corrupted?
Perhaps the Airbenders never really went extinct. Perhaps there were a bunch of Airbenders running around the Fire Nation, but because they grew up in the Fire Nation, none of them ever discovered their talents. Where did the Air Nomads come from anyway? Nomads have to start from somewhere.
I realize this is all just speculation, but it's fun to think about.
On a totally unrelated note, I have been rewatching Season One of A:TLA, and I just watched the Winter Solstice two-parter. This is the episode where Aang first enters the Spirit World and meets Roku. An interesting thing I observed that I never noticed the first time around is that when Aang enters the Spirit World, he meets up with Roku's dragon. The dragon and Aang fly to Roku's Island, but on the way they fly over General Iroh and a bunch of Earthbenders. Iroh sees them!
They're in the Spirit World; nobody should see them, but Iroh does.
It's a story thread I feel must have later been dropped, but I wonder what the original intention of that was.