Perhaps Naruto has some unresolved issues from his past that he doesn't want to burden his family with. He may think that leaving is the best way to deal with these issues on his own, without putting his family in the middle of his personal turmoil. He might go off on a solo adventure to find himself and come to terms with his inner demons.
In some fanfiction scenarios, there could be a prophecy or a special mission that only Naruto can fulfill in Kumo. This mission might be related to protecting the village from a great threat or uniting the ninja world in a new way. So, he leaves his family behind to pursue this greater good in Kumo. He may believe that by joining Kumo, he can have a more significant impact on the overall situation of the ninja world.
In some fanfictions, Harry might leave his family because of a prophecy or a special mission. He might feel that he has to go it alone in order to fulfill this prophecy or complete the mission successfully. His family might not understand or be able to support him in this, so he decides to leave. Also, there could be jealousy or resentment within the family that makes the environment unbearable for Harry.
One reason could be that Naruto might feel he has outgrown Jiraiya's teachings in the fanfiction. He may want to explore his own path independently. Another possibility is that Naruto gets involved in a mission or a situation where he has to part ways with Jiraiya for the greater good. For example, if there is a threat in a far - off land that only Naruto can handle, and Jiraiya has other commitments elsewhere.
One reason could be to seek more power. In some fanfictions, Naruto might feel that the Leaf doesn't offer enough resources or training for him to reach his full potential. So he decides to leave to find a place where he can grow stronger.
One possible reason could be a difference in their goals. Naruto is from the Hidden Leaf Village with a mission to protect it and promote peace there. Rias, on the other hand, has her own responsibilities in the Underworld. Their conflicting duties might lead to Naruto leaving.
One possible reason could be a misinterpretation of family dynamics. Maybe in the fanfiction, the author portrays Naruto's family in a way that he sees as unfair or unloving. For example, if they are shown as neglecting him or putting too much pressure on him.
In the fanfiction, it might be that Naruto had different ideals and goals from his family. His strong will to pursue his own path of becoming Hokage, for example, could clash with the family's expectations, causing him to reject them.
Perhaps there were secrets in the family that Naruto discovered. In fanfiction, it's common to introduce hidden family secrets. If Naruto found out something like his family having a connection to an old enemy or some dark past, he could be angry and snap at them.
One possible reason could be a misunderstanding. Perhaps Naruto misinterprets some of his family's actions or words. Maybe he thinks his family didn't care about him as much as he thought they should. In fanfiction, authors often play with the idea of miscommunication and how it can lead to strong negative emotions like hatred. Also, if there are other characters in the fanfiction who influence Naruto's view of his family, say by spreading false rumors or manipulating him, that could contribute to his hatred.
One reason could be a misunderstanding. Maybe there was a miscommunication between Harry and his friends, and he felt so hurt or angry that he decided to leave. For example, if his friends wrongly accused him of something he didn't do.