The writer might be trying to add some drama to the story. By making Aang hate Katara, it creates a conflict that can drive the plot forward. For instance, their friends might have to work hard to resolve this hatred and restore their friendship.
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.
Maybe in the fanfic, the writer wants to create a unique plot where Aang has some misunderstandings about Katara. For example, he might misinterpret her actions or words, leading to dislike.
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 in the fanfiction, Aang leaves Katara because of a misunderstanding. For example, he could misinterpret something Katara said or did regarding their relationship. This misunderstanding could lead him to believe that it's better for both of them if he goes away. He might think that he is protecting Katara in some way by leaving.
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.
One possible reason could be their different approaches to life post - war. Aang is the Avatar and has a whole world to look after, while Katara has her own tribe and personal goals. Their busy schedules might pull them apart. For example, Aang might be constantly traveling to different nations for diplomatic and Avatar - related duties, leaving Katara feeling lonely. Also, they might have different ideas on how to handle future threats. Aang may prefer a more pacifist approach which Katara might not always agree with, especially if it endangers the people she loves.
One possible storyline could be their adventures in rebuilding the world after the war. They might travel together, using Aang's bending skills and Katara's healing abilities to help people. For example, they could encounter a village suffering from a water shortage, and Katara uses her waterbending to find a new water source while Aang helps with the physical labor of constructing a new well.
Maybe in the fanfic, there could be a misunderstanding between them. For example, Ron might have accidentally broken Harry's trust in some way, like revealing a secret of Harry's that he wasn't supposed to.
Maybe in that fanfic, there are misunderstandings between Harry and the Weasleys. For example, they might have had an argument over something like magic use or family values that led Harry to dislike them.