One possible reason could be the constant scrutiny and pressure he faces in Konoha. Everyone has high expectations of him as the Jinchuriki of the Nine - Tails. He might feel suffocated by all the attention and the responsibility thrust upon him. Another reason could be the lack of true understanding from some of the villagers. Despite his efforts to protect Konoha, there are still those who look at him with distrust because of the Nine - Tails inside him.
Perhaps Naruto is tired of the way Konoha treats its shinobi. He might feel that the village is more concerned with using them as tools rather than caring for their well - being. For example, the way some missions are assigned without much regard for the safety of the shinobi. Also, his personal relationships might have hit a rough patch. If he has a falling out with some of his close friends or mentors, it could drive him to leave Konoha.
He could run to the Land of Waves. He has a connection there from his early mission. The people there were friendly to him, and he might find some peace and acceptance among them. It's a place where he could start anew, away from the chaos of Konoha.
In this fanfiction, Naruto might leave because of the political situation in Konoha. There could be factions that he doesn't agree with or that are causing unrest. Naruto, being someone who values peace and unity, might find it difficult to deal with the internal politics and decides to leave in order to find a more harmonious place or to figure out a solution away from the chaos.
Another reason could be for the sake of protecting the village. Maybe he discovers some sort of threat that he believes he can only face by leaving Konoha. He doesn't want the village to be in danger because of him, so he runs away to draw the danger away from Konoha.
One possible reason could be feeling ostracized. In the Naruto world, being different is often not easily accepted. As a female Naruto, she might face even more scrutiny and prejudice. Maybe she's tired of the constant expectations and the way she's treated in Konoha, so she decides to run away to find a place where she can be herself.
Another reason could be related to a prophecy or a hidden mission. They might have learned about a prophecy that required them to leave Konoha in order to fulfill it. This could involve finding a powerful artifact or learning a special technique that could only be achieved by being away from the village. Their departure would then be a sacrifice for the greater good of not just Konoha, but the entire ninja world.
One possible reason could be that he feels misunderstood by his family. Maybe they have different expectations of him and he can't bear the pressure, so he decides to run away to Kumo to start a new life.
It could also be due to a family issue. Maybe there is a conflict within the Uzumaki or Hyuga clans that makes them turn their backs on Konoha. For instance, if the village tries to interfere with the internal affairs of their clans in a negative way, they might decide not to offer their help anymore.
One possible reason could be past betrayals. Maybe he was mistreated by the village in some way, like being blamed for things he didn't do or not getting the support he needed when he was younger. So he feels no obligation to help.
It might be that his family is overprotective. Naruto is a very independent character. If his family in the fanfiction is constantly trying to hold him back from taking risks or going on adventures, he may feel suffocated. Running away would be his way of breaking free and exploring the world on his own terms.
One possible reason could be the constant feeling of being an outcast in the village. Despite his efforts, many villagers still look at him with suspicion because of the Nine - Tails inside him. He might feel that running away would give him a chance to find a place where he is truly accepted.