Well, it could be to show different relationship dynamics. In the original story, they are very close friends. By having Harry leave Hermione, it can show how their friendship can be strained and how they might react to such a big change. It can also lead to new plotlines like Hermione having to form new alliances or Harry finding new friends in different places.
Some fanfictions focus on 'Harry leaves Ron and Hermione' because it creates a lot of drama. The trio is so iconic in the original series that breaking them apart in a fan - created story can lead to all kinds of interesting plot developments. It also allows the author to explore the individual characters more deeply. Without Harry around, Ron and Hermione have to rely on their own strengths and face challenges alone, which can be a great character - building exercise. Another reason could be to introduce new relationships. Maybe Ron or Hermione forms a new alliance with another character in Harry's absence.
Some exist because it creates a really intense and dramatic conflict. Bellatrix is a very powerful and cruel character, and Hermione is a strong - willed heroine. Putting them in a situation where Bellatrix tries to break Hermione makes for a very engaging read.
They exist to add more drama and tension. Hermione is a beloved character, and putting her in a life - threatening situation amps up the emotional stakes for the readers.
Some 'Hermione breaks Harry' fanfics exist because it offers a new and different take on their relationship. The traditional view of their friendship is so strong that exploring a break in it creates a lot of drama and emotional tension, which is interesting for readers.
Some exist because fans like to explore different power dynamics in relationships. They want to see how Harry, as a central and powerful character, can influence Ginny's actions and decisions.
Some authors write these fanfictions to show different sides of the characters. For example, Snape's action of hitting Harry could be a result of his own inner turmoil, perhaps related to his loyalty to Dumbledore and his hatred for James Potter. By writing about this, they can show Snape in a new light. Also, it can show Harry's growth in dealing with such a difficult situation. He can either become bitter or try to understand Snape's motives, which are interesting character development arcs.
In some fanfictions, they leave because they might feel that Hogwarts has become a place full of danger and betrayal. For example, if there's a plot where Voldemort has infiltrated the school more deeply than expected, they would leave to protect themselves and find a new way to fight. Also, they could leave due to personal reasons like feeling ostracized by their peers or not agreeing with the way the teachers are handling things.
Some fanfictions like 'Harry Potter hides as a girl' exist because it's an interesting 'what - if' scenario. It allows authors to explore different aspects of the Harry Potter universe in a new light. They can play with the character dynamics and how the story would change if Harry had this different identity.
Some fans might feel that Dumbledore was too manipulative in the original series. They create these fanfictions to explore an alternative view where his actions are not excused.
One popular 'Harry Leaves Hermione' fanfiction might be 'The Unforeseen Path'. In it, Harry's decision to leave Hermione leads to a whole new set of adventures for both of them. Hermione, left on her own, starts to explore magic in a different way and Harry finds himself in a situation where he has to rely more on his instincts.