There are several possible reasons. For starters, some writers might want to create a contrast to the original Naruto story. In the original, Naruto is mainly driven by the lack of recognition from the village. By adding this abuse element, they can show a different source of his pain and motivation. Also, it could be a way to experiment with different character developments. Maybe they think that Naruto could have had a more complex personality if he had to deal with abuse from his father. Additionally, it might be just for the shock value, to attract readers who are interested in darker and more complex storylines.
One reason could be for shock value. Writers might want to create a story that stands out from the typical Naruto fanfictions. By introducing such a dark and unexpected element like abuse, they can grab readers' attention. Another reason could be to explore a different side of Sakura's character. Maybe in this fanfiction, they want to show a corrupted or extremely stressed version of Sakura who takes out her frustrations on Naruto.
In fanfiction, Naruto could be disowned by his father because of secrets. Suppose his father had some dark secrets in his past that Naruto unknowingly got involved with. His father might disown Naruto to keep those secrets buried. Or perhaps Naruto's father was under the influence of some dark magic or mind - control that made him act against his normal feelings for Naruto. It could also be a case of a prophecy that led his father to believe Naruto was a threat to the village in the future, even if there was no real evidence yet.
Some authors might write it to add a new and unexpected twist to the Percy Jackson story. They may want to see how the other characters would react if such a thing happened. Also, it could be an attempt to explore the psychological impact on Percy in a very different and extreme situation, though this is a rather controversial approach.
One reason could be to explore a darker side of crossovers. Writers might want to see how a typically positive and powerful character like Naruto would react in a situation where he's being abused. It can create a lot of drama and tension in the story.
One possible reason could be that Naruto discovers some dark family secret that his parents want to keep hidden. For example, if there was a connection between the Uzumaki clan and some forbidden jutsu, and they fear Naruto might be exploited because of it. So they banish him to protect the clan's reputation and his own safety.
One possible reason could be that she is influenced by an enemy. Maybe the enemy has some sort of leverage over her or has brainwashed her into believing that Naruto is a threat. Another reason might be a misunderstanding. For example, she could misinterpret Naruto's actions as being unfaithful or harmful when in reality he has good intentions. It could also be due to her own insecurities. If she feels overshadowed by Naruto's popularity and power, she might betray him to assert her own identity.
One reason could be to explore complex psychological and emotional states. By making Percy abused and mute, the writer can delve into how a character copes with trauma and isolation. However, this is a very dark approach.
One possible reason could be that Naruto feels abandoned. His father was not around during his childhood, and he had to face a lot of hardships alone. Growing up without a father figure, he might have developed resentment.
Perhaps the father is under the influence of some external force. In fanfiction, it could be that he is being blackmailed or controlled by a dark wizard. As a result, he has to reject Harry against his true feelings. Also, it could be that the father has made some big mistakes in his life and he doesn't want Harry to be associated with him because he thinks it will bring harm to Harry.
It could be that his sister has some special abilities or talents that the family or the village values more. For example, if she has a very rare and powerful bloodline limit, everyone's attention would be drawn to her. In a ninja world, power and potential are highly regarded, so Naruto might be overlooked because his sister seems to have more of that 'special something' at first glance.