I could see Aang simply refusing to do it. He took Ozai's firebending away because it was literally the only option he could find that didn't result in taking his life. In this world, bending is a part of a person's nature, be it genetic or spiritual, and I don't even think the Avatar (at least, not Aang's Avatar) has the authority to go around making new benders just because he can.
Not "just because he can." Because he had to
for the survival of his people and the future balance of the entire world. That balance only survives if airbenders survive, and they're a critically endangered breed.
'Future balance of the entire world'? Profound sounding words with no clear meaning. Care to define what this means, exactly? And what the consequences are of no airbenders existing? Beyond unspecified 'bad things'?
Aang's children are not 'his people' if they can't airbend? That's cold.
I guess that explains why Aang had no time for them - not even bothering to take them along with his favourite for a vacation.
After all, his favourite would be no less of an airbender if Aang would have bestowed less paternal love on him and more on his siblings.
I see you're still claiming your vision of the avatar-verse (consisting mainly in avoiding any 'western' influence - and you're using an over-simplified caricature as western mentality) is the only valid one. With few arguments that don't consist of hand-waving, that is.
Teaching a non-bender to bend may be beyond the avatar's possibilities - but a non-bender becoming a bender may be possible, if the relevant spirit approves the application, as happened to the first avatar. etc, etc.