Okay, so there is a lot to address here, and honestly, I had a hard time knowing where to begin because wow...
First of all, all I see here is how Aang feels. Why is it that whenever anyone discusses Kataang, it's all about Aang and his feelings? Even in the show, it's only ever really shown from his side and what he wants, not Katara. Does Katara have a say? Is katara's only purpose in life to be an airbending baby maker? Does Katara's very existence hinge other being Aang's girlfriend? What about how she feels? What about what she wants? What about her happiness? Or is Aang's happiness all that matters, and Katara should just deal with it?
Now for the culture argument. I want to point out that no genocide is ever really successful in wiping out an entire group of people. Air nomads are literally nomads meaning they move around a lot. It's in the name. This means that there are most definitely other airbenders out there hiding too afraid to come out until after the war is over. But, let's say we go with all the airbenders are dead narrative, then it makes even less sense for Kataang to happen because that means the entirety of the future of one nation depends on how many airbending babies Katara pops out. So this plan is stupid. But Aang is 12, so this plan might make sense in his brain. However, as he grows up, he would come to the obvious realization that if he wants to bring back the airbenders he needs to make a lot of babies with a lot of different women, so monogamy makes no sense and Katara would likely not be down for her boyfriend/husband "planting his seed" in every willing woman given that she's from the water tribes where family units are central to her culture.
Another thing Aang is 12, he'll survive not getting his first crush to date him. We all have at some point or another experienced heartbreak over someone who didn't like us back. We get sad for a bit and then move on. That's life. I also speak from experience as someone who was in a relatioship at 12 that no one should date at that age, and it is extremely unlikely you will end up together in the end. My relationship lasted a grand total of three months, and it is by far the dumbest decision I ever made. I was an idiot, he was an idiot, we were both idiots not ready for a relationship like all 12-year-olds are. So the idea that Aang and Katara would even last beyond a few months is ridiculous and unrealistic. In fact, the idea of me still being with that same person makes me visibly cringe because it was very unhealthy for a multitude of reasons I am still recovering from emotionally. If I was still with him, I would not be in the amazing relationship I am in now. Every relationship contains lessons that help us grow, and we keep with us when entering another relatioship. Sometimes we grow in relationships, and we realize we aren't right for each other, and that's okay. That's life. It goes on. That said, I know some people do get together at that young age and end up happily married. I know two people from my high school personally who got together at 14, got married a year after high school, and just had their first baby. But, the likelihood of that happening is extremely slim. On that note, relationships are hard work and require a certain amount of maturity. The maturity gap between Katara and Aang is vast, with Katara having the mental age of at least 25 and Aang still having the mental age of 12. It's unbalanced.
Furthermore, the idea that Aang's happiness depends on Katara being with him means that his other relationships pale in comparison to his crush. It also makes him kind of pathetic if he's gonna end up "dying of depression next to Monk Gyatso's skeleton" You so eloquently put it if he doesn't get with Katara. Also, it makes the connections he has with the others in the gaang worthless, especially his friendship with Katara. Because that means Aang doesn't really care about being in Katara's life if it isn't in the way, he wants it. It makes their friendship conditional, with no value because there was always an ulterior motive, and in his mind, it was always just a means to an end. That just makes him a selfish dick.
The argument that Aang is alone without Katara is dumb. He has Toph, Sokka, Suki, Katara, and Zuko. Plus, I don't know if you know this, but you can make more friends than the ones you made in your teens. You don't just stop making friends after a certain age. You continue to meet people and form connections. And given how charismatic Aang is, he's gonna make a ton of more friends and have the opportunity to form new bonds, including romantic ones as well. Furthermore, as a 12-year-old, Aang should be allowed to expand his circle around the world because he is the Avatar, and he is going to need to make new friends because his main ones aren't always gonna be available. Moreover, the idea of Aang being alone without Katara implies that Aang will only ever have katara at his side and cut ties with everyone else and expect her to do the same.
The idea that to have a happy ending Aang must get into a romantic relationship makes no sense. What about Toph? She didn't have a romantic relationship in the end, and she was perfectly happy. Because she's 12 and knows life doesn't depend on being in a romantic relationship all of the time. Furthermore, on the topic of Toph, I would like to bring up another argument you made about Aang not developing other potential love interests because he does. Toph is one of them. In season 2, we are introduced to Toph by way of a vision that is pretty romantic and used as a common trope in both modern and ancient literature. We also see their compatibility in the way they are naturally with each other. Aang wants to be a kid and have fun. With Toph, he can do that. But, he also has someone who won't let him shirk his responsibilities when they matter most. They match and balance each other in their personalities, and we see some possible romantic development in both seasons 2 and 3. Another potential love interest we are introduced to is On Ji in season 3, episode 2, "The Headband," where we see her have an obvious crush on "Kuzon" (Aang.) But what's appealing about this pairing is that she doesn't know he's the Avatar, meaning she genuinely likes him for who he is without the glitz and glamour. Which I personally find adorable. Side note, I am aware that "the headband" is supposed to be a hallmark "kataang" episode. However, I would be remiss if I did not point out that this episode actually proves the "Katara doesn't treat/see Aang as a son" argument wrong. Because in that same episode, before the dance scene, Katara literally pretends to be Aang's pregnant mother.
Moreover, Aang is not without purpose outside of being the Avatar. The very notion is ridiculous. Yes, he has to rebuild his nation, he's a symbol of peace, but he is also a person who wants friends and a family, which he has found in the gaang. He also has the purpose of being a kid, which now he can be contingent on him meeting his obligations as the Avatar.
Lastly, I would like to reiterate that not only is Aang 12 and a child, the rest of the gaang are too. Granted, Aang is by far the least mature as he grew up in a time without war and has not had to a childhood with the effects of war the other members of the gaang have throughout their lives. But, in regards to Zutara, I personally do not see them getting together immediately after the war, they both have responsibilities to their respective nations. Katara, in rebuilding her tribe and Zuko in redefining the Fire Nation. Therefore, it would take years just to find some stability. But that doesn't mean they don't keep in touch. We know out of everyone in the gang, Zuko and Katara are closest to each other. They both share a bond with each other no one can match. You might be tempted to argue that Aang is closest to Katara and her best friend, but that is pretty debatable because both Aang and Katara have more positive interactions with other gaang members than they do with each other. For instance, if I had to choose out of everyone in the Gaang who Katara's best friend was, aside from Zuko, I would be tied between Toph and Suki. But, given Suki is not part of the gaang long enough to really expand on her relationships with the others beside Sokka, Toph would be my answer. That said, after the war, everyone would keep in touch with each other. They are all friends for life regardless of romantic relationships. Therefore, the gaang would most definitely be keeping in touch with each other, not just Zuko and Katara. But over the years as they grow, I can see Zuko and Katara growing closer and building on that strong foundation to form a strong and healthy romantic relatioship.
