One possible reason could be his pursuit of power. Naruto might feel that he can't achieve his full potential in the current place and needs to go on a journey to become stronger. Another reason could be to fulfill a promise or a sense of destiny. He may have a calling that he believes can only be answered by leaving.
Naruto could be driven by a thirst for knowledge. He might have heard about certain jutsus or powers that are only available in other places. He also may feel restricted by the village's rules and regulations and leaving gives him the freedom to explore and grow in ways that are not possible if he stays.
One reason could be that the writer has outgrown the Naruto fandom. As people grow and change, their interests shift. They might find other anime or stories more appealing now.
Sasuke could end up in a very dark place. He might get involved with some really bad characters or groups that could lead him further astray from his original path. He could lose his sense of self and become consumed by revenge or whatever negative emotions are driving him to leave in the first place.
Perhaps in this fanfiction, Yugito was involved in a political intrigue within the Hidden Cloud Village. Naruto, being an outsider with influence, banished her to maintain peace between different factions. Yugito's actions or associations might have been such that it was the only way Naruto saw to prevent a civil war or major conflict in the village.
In some fanfics, Naruto may ignore Sakura because he has his own emotional traumas to deal with. He might find Sakura's presence a reminder of things he wants to forget or move past, so he chooses to keep his distance. For example, if he has faced a lot of rejection from her in the past, he could decide to just cut her out of his life.
It could also be due to Naruto's outcast status in the village initially. Danzo might think that banishing Naruto would be an easy decision as Naruto doesn't have many supporters at first. He might believe that by getting rid of Naruto, he can maintain the status quo in the village and prevent any disruptions that Naruto's presence might cause in the future.
One reason could be to create shock value. By having two main characters kill another important character, it grabs the reader's attention. Another could be to explore a very dark and alternate version of the Naruto universe. Maybe the writer wants to show how far characters could be pushed in extreme situations.
One possible reason could be Naruto's over - eagerness to prove himself. Naruto is always trying to show that he's strong and capable, and sometimes this might come across as him not considering Sakura's feelings or opinions. For example, he might rush into a dangerous situation without consulting her, which could lead to an argument as Sakura is also a capable ninja who wants to be part of the decision - making process.
One possible reason could be that Naruto has grown tired of Sakura's violent tendencies towards him. He might have reached a point where he decides to stand up for himself and not tolerate being hit any longer.
One possible reason could be that Naruto feels he has learned all he can from Jiraiya at that point. Maybe he wants to explore his own path independently and test his skills without Jiraiya's constant guidance.
It could lead to a major shift in the relationship dynamics. For example, it might create a rift between Sakura and Naruto that will be hard to mend. This can add a lot of emotional depth and conflict to the story.