With the nations intertwining, the Avatars after Korra will be difficult to track down.
Example: Say the next Avatar is born from an earthbending mother and a firebending father and is the youngest of 3. His or her older siblings all came out to be firebenders so the child decides to try firebending and BOOM! fire comes out. So as the world awaits for the next Avatar to be reborn, they are oblivious to the fact that the child they deemed a firebender is actually naturally an earthbender. Make sense? Then years down the line when their life is endanger and they fall off a cliff running away, they airbend out of nowhere and just be like, "I’m the Avatar?"
It’s evident that the Avatars don’t have to learn the elements according to the cycle (as seen in ATLA when Aang learns firebending before earth) so what if any of the future Avatars learns another bending element other than their natural element?
So an Avatar could live their entire lives believing their a certain bender when in actuality their natural bending ability is another element. Does that make sense?