One possible reason could be his strong moral and spiritual beliefs. Aang is a very spiritual character, and his culture and upbringing might have instilled in him the value of all life. Killing goes against his principles of peace and harmony.
It could also be due to the fear of the consequences. In the Avatar universe, taking a life might have unforeseen spiritual or karmic ramifications. Aang, being the Avatar, has a connection to the spiritual world and might be aware of these potential negative effects. So, to protect himself and the balance of the world, he refuses to kill.
His refusal can also build tension. Other characters might not understand his stance, or they might be frustrated by it. This can create conflict within the story, both between Aang and his allies, and between Aang and his enemies who might not expect his mercy. For example, some of his friends might think that killing a particularly dangerous enemy is the only way to ensure safety, but Aang stands firm, causing a rift in their relationships. This adds an extra layer of drama to the plot.
It might be popular because it stays true to the essence of Aang's character. Fans who love the original show and Aang's pacifist nature would be drawn to it as it explores that side of him further.
The 'aang doesn't kill fanfiction' could be unique in terms of the relationships it portrays. Since Aang doesn't kill, it might show how his friends and foes react to this. Maybe some friends admire his resolve, while foes might underestimate him at first. This can lead to complex relationship dynamics that are different from the typical action - based stories where killing is more common.
In some fanfictions, it might be part of a plot to create tension. The writer wants to show that their relationship isn't all smooth sailing. So, Katara hurting Aang can be a device to make the story more interesting. It could also be related to different character arcs, like Katara trying to become more independent and in the process, her actions accidentally harm Aang.
It could also be that Katara was being overly critical of Aang's progress in learning new skills. Aang might feel frustrated and angry if Katara is constantly pointing out his flaws instead of being more encouraging, especially if he's already trying his best.
Perhaps Aang has a vision of the future where Katara is happier without him. So, in the fanfiction, he decides to respect that future and let her go.
Maybe Zuko has his own goals that don't involve chasing Aang. He could be more focused on redeeming himself in other ways, like helping the Fire Nation people or learning about different cultures without the need to pursue Aang.
One possible reason could be their fundamental differences in personality. Harry is brave and impulsive at times, while Snape is often brooding and secretive. Their past interactions, which were mostly filled with animosity and mistrust, could also be a factor. Harry might not be able to see Snape in a romantic light due to all the times Snape was cruel to him in school.
One possible reason could be unrequited love. Maybe he has strong feelings for a character who doesn't feel the same way about him. For example, if he has a crush on Katara but she falls for someone else. Another reason might be the loss of a close friend. If Zuko or Toph were to sacrifice themselves to save the world, Aang would be heartbroken as they are his dear friends. Also, if he fails in his duty as the Avatar to protect the world from a great threat, he could feel a deep sense of heartbreak and failure.
It could be that the fanfiction writers wanted to explore a more complex and darker side of relationships. They might have written scenarios where Aang hates Katara due to jealousy, perhaps over her friendship with another character. This would add an interesting twist to the original relationship dynamics between them.