Perhaps in this fanfiction, Yugito was involved in a political intrigue within the Hidden Cloud Village. Naruto, being an outsider with influence, banished her to maintain peace between different factions. Yugito's actions or associations might have been such that it was the only way Naruto saw to prevent a civil war or major conflict in the village.
It could also be due to Naruto's outcast status in the village initially. Danzo might think that banishing Naruto would be an easy decision as Naruto doesn't have many supporters at first. He might believe that by getting rid of Naruto, he can maintain the status quo in the village and prevent any disruptions that Naruto's presence might cause in the future.
One possible reason could be a misunderstanding. Maybe some false information is spread about Hinata that makes Naruto believe she has done something unforgivable. For example, if there's a plot to frame her for treason against the village. Another reason might be for her own safety. Naruto might think that banishing her would keep her away from a greater danger that is targeting her because of her relation to him.
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.
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 that Sasuke is highly regarded in the village. His skills and potential are seen as valuable assets. So, when Naruto hurts him, it might be seen as a major threat to the village's future strength. Also, the village elders may be overreacting due to their long - standing distrust of Naruto because of the Nine - Tails within him.
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.
One possible reason could be his pursuit of power. Naruto might feel that he can't achieve his full potential in the current place and needs to go on a journey to become stronger. Another reason could be to fulfill a promise or a sense of destiny. He may have a calling that he believes can only be answered by leaving.
The plot could be that Yugito is jealous of Naruto's friendship with Hinata. Hinata is always shyly helping Naruto and Yugito misinterprets their interactions as something more. So, she tries to get closer to Naruto by challenging Hinata to various tasks. But Naruto is oblivious to Yugito's feelings at first, which adds to the drama.
One thing that could happen is that they would have to deal with the cultural differences between the Leaf and Cloud villages. Naruto might introduce Yugito to all the unique things in the Leaf Village like Ichiraku Ramen. And Yugito could share her Cloud Village traditions with Naruto. Their relationship could also inspire other characters to look beyond village boundaries when it comes to love.
Another reason could be that fanfiction has the potential to create alternate universes that deviate too much from the canon. Naruto fans who are very attached to the official story might fear that these extreme deviations in fanfiction could overshadow or misinterpret the original narrative. For instance, a fanfiction that makes Naruto into a villain instead of a hero might be feared by those who cherish the original story.