One possible reason could be a search for power. The character might feel that Konoha doesn't offer enough resources or training opportunities for them to fully explore the potential of the Rinnegan. They may believe that there are other places in the Naruto world where they can gain more knowledge and power related to their special eye.
Perhaps they are being persecuted or misunderstood in Konoha. People in the village might be afraid of the power of the Rinnegan and treat the character differently. This could lead to the character feeling alienated and deciding to leave in search of a place where they will be accepted for who they are and what they can do.
Madara Uchiha could also be the main character in this type of fanfiction. Although in the original story his actions are mostly related to his grand plans for the Ninja world, in a fanfiction, it could be explored what would happen if he decided to leave Konoha specifically because of his Rinnegan and the power it holds.
It could be Sasuke. If he were to turn to the dark side in the fanfiction, his Rinnegan powers could be used to destroy Konoha. He has the potential to be extremely powerful and if he has some sort of vendetta or is influenced by some dark force, he might take his anger out on the village.
Well, it could be that the wielder of the Rinnegan has the ability to control powerful forces like the Six Paths techniques. They might use the Deva Path to create massive gravitational pulls that start to crumble the buildings in Konoha. Or perhaps with the Preta Path, they could drain the chakra from all the ninja in Konoha, leaving them defenseless against further attacks, which eventually leads to the destruction of the village.
Another reason could be that he witnesses corruption or unfairness within the village leadership. If he sees that the system is rigged against certain groups, including himself or his friends, he could grow to hate Konoha and decide to leave.
Another reason might be that Naruto is seeking a greater purpose. In the fanfiction, his 'godlike' state could make him realize that there are threats out there that he can't face while being tied to Konoha. He may leave to protect the world on a larger scale, believing that he can be more effective away from the village.
Maybe the main antagonist is a powerful figure within Konoha who doesn't want Naruto to leave and become an independent force. He secretly sends assassins after Naruto. When Temari gets involved, this figure also targets her. His actions are driven by his thirst for power and fear of Naruto's potential outside of Konoha's control.
The elders of Konoha could be the main antagonists. They might be the ones who have been neglecting Naruto all along, and when he leaves, they see him as a threat to Konoha's stability.
In this fanfiction, perhaps Naruto leaves Konoha because he's being framed for something he didn't do. As he flees, he encounters Temari. Temari, being a strong - willed character, might be intrigued by Naruto's situation. She decides to help him clear his name. During this process, they travel to different places, learn about each other's pasts and dreams, and their relationship evolves from distrust to mutual respect and then to a deeper connection.
Well, perhaps Naruto is neglected in Konoha because of his past or the way he behaves. He leaves and stumbles upon a hidden village that has different values. There, he is welcomed and trained further. As he grows stronger, Konoha realizes their mistake. Naruto has to decide whether to forgive Konoha and return or stay in his new - found home where he is respected.
Perhaps Naruto leaves because he wants his child to grow up away from the pressures and expectations that come with being in Konoha. He might feel that the outside world can offer a more peaceful and normal upbringing for his family.