Okay bitches, it's time for an explanation (of how I see it)!
First things first: By "every airbender avatars" I mean every future avatar that was a native/born an airbender, like Korra is native waterbender. Of course every Avatar will be able to airbend sooner or later. Maybe I should have been more specific about that...

Second: We all pretty much know that bending is somehow genetically stored in the genes. That means that Kya and Bumi MAY be able to "produce" airbenders, just like Tenzin, though it isn't guarantied. But that STILL means that those airbenders would be descendants of Aang since Kya and Bumi are Aang's children (duh...).

"Yeah, well, uh... How come Mako and Bolin are brothers but bend different elements?" Simple! Their father was an earthbender and their mother a firebender, they just got different pair/sets/combinations of genes from their parents, just like Aang and Katara's kids did (or however you say it in english). That is also why there were only earthbenders born in the Earth Kingdom, only waterbenders born in the Water Tribe, and so on, before Korra's time. The different peoples of the earth didn't interact with each other nearly as much as they do now when the industrial revolution is in full swing and after The United Republic of Nations is founded, a place were benders and non-benders from all nations live and interact with eachother every day. Of course there might have been " *********** " couples before and during Aang's time, but they were not nearly as common as they are now.

"But if someone very spiritual is born, doesn't he become an airbender, since they are spiritually stronger than the other benders and that is a requirement for being an airbender?"
Well, I too have heard somewhere that airbenders are all supposed to be very spiritually connected when thay are born, or something like that, otherwise they wouldn't turn out to be an airbender. But by just being very spiritual doesn't mean the child becomes an airbender. They have to have the right genes too! "And how do you know that?" you may may ask. Well, where the **** were all the newborn little airbenders during the hundred year war, then?! We know that unkle Iroh was so spiritually wise and strong that he could even see and communicate with the spirits, but he was still just a firebender because he lacked the right genes. Not even the people living with Tenzin on Airtemple Island and in the other airtemples in the world are, or will ever be, airbenders. They are just acolytes: people who practise the spiritual way and living of the airbenders.
There's a reason Aang was ******* called "The Last Airbender", because that's exactly what he was! He was the only one of his kind, the last one, meaning he was the only one that carried the right genes to produce an airbender kid, which means that Aang will be every future native airbender's common ancestor!

This glorious ************ will not only be remembered as one of the most successful avatars, "The boy in the iceberg", "The last airbender",and "the founder of Republic City". He will also forever be the FATHER OF ALL NEW AIRBENDERS!!!!!
Aang is so ******* ******** !

Tl;dr: Just read it, you lazy bum!