Maybe he discovers some dark secrets about his family. For example, if his parents were involved in something he considered unethical in the fanfic's plot. It could be that they were part of a group with a hidden and evil agenda.
He might run away because he feels a lack of understanding from them. In the fanfic, his parents could be so preoccupied with their own matters, like protecting the village in a way that makes Naruto feel neglected. He may think that by running away, he can find his own path and purpose without their overshadowing influence.
One possible reason could be that he discovers a dark secret about his family. Maybe he finds out that his parents were involved in something he can't accept, like a deal with a dangerous enemy. So, to avoid being associated with it, he runs away.
One possible reason could be the pressure he feels in Konoha. Naruto has always been an outcast in the village to some extent. People used to fear and shun him because of the Nine - Tails inside him. He might feel that running away would be a way to escape all that negative attention and find a place where he could be himself without the constant reminder of his 'otherness'.
One reason could be a misunderstanding within the family. Maybe Naruto feels that his family doesn't understand his dreams or his ninja way. For example, if his family has different expectations for him, like wanting him to focus on clan duties instead of his own adventures and growth as a ninja. Another reason might be that he discovers a dark secret about his family and can't bear to be associated with it, so he runs away to find his own truth.
Another reason could be that Naruto is being targeted by a powerful enemy within Konoha. He doesn't want to put his friends and the village at risk, so he runs away to draw the enemy away from Konoha. This would show his selfless side, sacrificing his own safety for the greater good of the village.
One possible reason could be that he discovers a dark secret about his family. Maybe he finds out that his parents were involved in something dangerous or unethical in the wizarding world, and he can't bear to be associated with it, so he runs away.
One possible reason could be that Naruto feels like an outcast. Despite being part of Team 7, he might think that Sasuke and Sakura are closer to each other. Naruto has always been alone before joining the team, and if he feels excluded again, it could lead to him running away. Also, he might be afraid of endangering his teammates. Naruto often attracts trouble because of the Nine - Tails inside him. He may think that by leaving, he can keep his friends safe from the enemies who are after him for the power of the Nine - Tails.
Another possibility is that they were deceived. There could be a villain who tricked them into believing that giving away Naruto's birthright was the right thing to do. They might have been misinformed about the consequences and the true nature of what they were giving up.
He could be running away because of false accusations within the family. Just like how he was wrongly judged by the village in the original story, if his family wrongly blames him for something he didn't do, like stealing a family heirloom or causing harm to a family member accidentally, he might choose to run away rather than face their unjust wrath.
Maybe he's tired of his restricted life. In the story, he might be feeling suffocated by all the rules and expectations placed on him, so he decides to run away to find freedom.
One possible reason could be Naruto's internal conflict. In yaoi fanfiction, the relationship is often complex. Naruto might be running away because he's confused about his feelings for Sasuke. He may not be ready to face the intensity of the relationship that the yaoi concept implies.