Maybe he runs away to find out more about his parents. He could be frustrated with the lack of information in the village and decides to take matters into his own hands. He might believe that by leaving, he can uncover the truth about his family on his own journey.
One reason could be that Naruto feels constantly ostracized in the village despite his efforts to be accepted. In some fanfictions, he might run away because he discovers a dark secret about the village elders that makes him lose faith in the place he once called home.
One possible reason could be that he felt misunderstood. In the village, Naruto was often ostracized despite his good intentions. Maybe he got tired of trying to prove himself and just ran away. For example, when the villagers still saw him as the nine - tailed fox rather than Naruto, it might have pushed him to leave.
One possible reason could be that he felt misunderstood. In the Naruto series, Naruto was often an outcast in the village. Maybe in this fanfiction, his family didn't understand his dreams or his need for recognition. So he ran away to find a place where he could be himself.
It could be due to a family secret. Suppose his family was involved in something dangerous or unethical that Naruto couldn't bear. For example, if they had ties to an enemy organization that Naruto discovered. He is a character with a strong sense of justice, so running away would be a way to distance himself from their actions and figure out how to deal with the situation on his own.
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.
Well, it could be that Naruto, being the brother of Lucy, ran away perhaps due to some sort of family conflict. Maybe he didn't agree with the way their family was handling things, like their training methods or the expectations placed on them. Another possibility is that he ran away to protect Lucy from some unseen danger. He might have found out something dangerous was coming their way and thought it was better to lead the threat away from her by running off.
Perhaps he is running away from danger in Konoha. There could be a threat that he believes he can't face directly in the village, and he thinks Whirlpool might offer some sort of sanctuary or a way to become stronger in relative safety.
One possible reason could be that his godlike powers were seen as a threat. The village elders or some powerful forces might have feared that Naruto, with his immense strength, could lose control and cause destruction. So, they decided to seal him away to protect the village and the world at large.
Maybe he's running away from some kind of danger or threat in his home village. There could be a group or an individual after him, and he believes that Kumo would be a safer place. It could also be related to some political unrest in Konoha that makes him feel like he has to leave, and Kumo seems like a viable option.
One possible reason could be fear. Minato is a very powerful shinobi, and Naruto might be scared of him for some reason. Maybe he has heard stories about Minato that made him afraid.
Perhaps Naruto realizes that forcing Sasuke to stay would only create more resentment. By letting him escape, he hopes that Sasuke will have time to think and eventually return willingly. In the fanfiction, it could also be that Naruto has some sort of premonition that Sasuke's escape is necessary for a greater good in the future, like saving the village or the ninja world in a different way than if he were held back.