Maybe he runs away to protect the village. He could discover some sort of threat that is targeting him but would endanger the village if he stays. So, in his selfless way, he decides to leave to draw the danger away from his home and friends.
One possible reason could be that he feels misunderstood. In the village, he has always faced discrimination because of the Nine - Tails inside him. People often look at him with fear and suspicion. So, he might run away to find a place where he can be accepted for who he is.
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 that he feels misunderstood. In the Naruto series, Naruto often faced isolation and judgment from the villagers. If this extended to his family in the fanfiction, he might run away to find a place where he is accepted. Maybe he discovers some dark secrets about his family that make him feel unsafe or betrayed.
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.
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 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.
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.
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.
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.
It could also be due to a sense of confusion. Naruto suddenly finds himself in this unexpected situation. He doesn't know how to deal with it and feels that running away is the easiest way at the moment to escape from all the chaos and figure things out on his own.
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.