Late fan to everything

From expanded lore, we know that their 'spritual purity' for a lack of a better phrase meant that all air nomads were airbenders. Now, I already believe this to be a load of shit, but I'll assume that this is correct. Because we know from the Kyoshi books that Avatar Kyoshi's mother was an airbender who lost touch with much of her bending ability as she drifted from that spiritual path.

This is actually a phenomenon which is evidenced in the original ATLA show through Zuko. Initially, the core of his firebending was tied to his anger and hatred. As he moved past those, his flame weakened until he could rekindle it by connecting spiritually to his bending. So we know that benders can fade. We also know that non-benders can have kids who bend (Katara's parents for example).

This means that Aang assumes that it is impossible for anyone who is a non-bender currently to become or give birth to an airbender. As such, he doesn't seek to really train non-benders in Air Nomad culture. They are excluded from his efforts to continue his culture.

That is a profound prejudice. It's as profound as any nationalistic animosity. And Aang keeps to it. In TLOK his neglect of his children is profound - but it is Bumi specifically who is most hurt because he wasn't even a bender. And we see with Tenzin how Aang's obsession separated Tenzin emotionally from his family, pulling him out of the community of his own blood.

And we know there are non-bending acolytes. But the fact is that Aang always prioritizes the potential for an airbender over any non-bender.

So we have an Aang who is clearly prejudiced - who isn't focused on his culture, but specifically on an ethno-cultural articulation of it focused on airbenders rather than the broader air nomad philosophy.

Cool. If the story ever acknowledged it meaningfully, it would have made a really good narrative thread. But oh well!

But the other aspect of it is this: Aang sees himself as the Last Airbender. He sees himself at the end of the line. Which is at odds with the Avatar Cycle. And he should know this. So his belief in himself as the last guy who will airbend natively (as opposed to airbending as an avatar) means he doesn't believe that an airbender can be born to people who are non-benders. He doesn't believe that, when the cycle comes back round, it will find a way.

And for someone whose people are defined by their spiritual purity and are in lore articulated as having superiority over the other races nations in their connection to the spiritual, it is a deep lack of faith in the very mechanisms of the world.