One possible reason could be for Naruto's safety. Maybe she has to go on a dangerous mission and believes Konoha is the safest place for him. Another reason might be that she is forced by some powerful enemies who threaten Naruto's life if she doesn't leave him in Konoha.
It could create a sense of unease and fear. Naruto is a very powerful ninja, and if he turns dark and leaves, the villagers would be worried about him coming back as an enemy. It might also lead to a decrease in morale among the ninjas who once considered Naruto a friend or ally.
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.
One possible reason could be that he feels ostracized. In the Naruto series, Naruto was often treated as an outcast in Konoha in his early days. If he turns dark, this feeling might be magnified to the point where he decides to leave. He may think that Konoha doesn't really accept him and that he can find a better place or a new purpose elsewhere.
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.
He may face some difficulties in understanding his identity. Since Kushina is not there to tell him about his family background, he might have to discover it on his own, which could lead to some misunderstandings and challenges. Also, his relationships with other characters in Konoha could be different. For example, he might seek more father - like figures among the villagers, which could change the dynamics of his interactions with them.
Naruto's character would likely become more independent. Away from Konoha, he has to rely on himself and the female Kyuubi. He might become more cautious as he no longer has the safety net of his village. His view of friendship could also change. Since he's with the female Kyuubi, he might develop a deeper understanding of different beings and their motives.
Well, it could go in many directions. Maybe Naruto starts a quest to find a way to bring peace to the ninja world on his own terms. Or perhaps he has to deal with the consequences of leaving Konoha and tries to prove his intentions were pure.
Well, it could be that Naruto leaves because he realizes that Konoha's ways don't align with his ideals anymore after that intense mission. Or perhaps there are hidden threats that force him to go away for the greater good.