Naruto's relationship with the other Jinchuriki would likely change. If Naruto hates Yondaime, he might see the system of creating Jinchuriki in a more negative light. He could try to convince the other Jinchuriki that they have been wronged by the villages and their leaders like Yondaime. This could lead to either a united front among the Jinchuriki against the villages or some internal conflict among them depending on their individual beliefs.
Well, in the original, Yondaime was Naruto's father and a hero. But in this fanfiction, if Naruto hates him, it completely flips the relationship. It makes Yondaime more of a villain - like figure in Naruto's eyes.
Perhaps Naruto blames Yondaime for his mother's death. In the original story, Kushina died while giving birth to Naruto, and if in this fanfiction there are some hidden motives or actions of Yondaime related to that event, Naruto could hold a grudge. Like if he found out that Yondaime could have saved Kushina but chose not to for some greater good that Naruto doesn't understand or accept.
One possible reason could be a misinformation plot. Maybe in the fanfiction, Naruto has been led to believe that Yondaime had some dark secrets or did something bad to his family that he discovers later in the story.
Well, in some fanfictions, the idea of Naruto hating Kushina might be a very dark and twisted take. Maybe the author wants to explore extreme emotional scenarios. As for Naruto loving in this context, it could be about different kinds of love - like platonic love for his friends or a romantic love for a character. Fanfiction often takes liberties with the original story's relationships.
Maybe she was forced to for some dark reason. Perhaps there was a threat to Naruto's life if she stayed, and she thought leaving was the only way to protect him.
In a fanfiction like this, Fem Kyuubi's love for Naruto could be a deep, almost possessive kind of love. Since Naruto has always been different from others in how he treats the Kyuubi. As for hating Kushina, it could be that Fem Kyuubi blames Kushina for separating her from the world for so long. And this could lead to a lot of internal conflict within Naruto, who loves his mother but also has a bond with Fem Kyuubi.
Maybe in this fanfiction, there was some sort of dark magic or mind - control involved. Kushina could have been unknowingly influenced to do something that deeply hurt Naruto. Like she was made to abandon Naruto at a crucial moment in his life, which then led to his hatred.
One possible reason could be a misinterpretation of Kushina's actions. Maybe some events were presented in a way that made Naruto think she was neglectful or had made bad decisions in the past.
Maybe in this fanfic, the author wants to create a unique and dark plot. Kushina might be made to hate Naruto due to some kind of misunderstanding. For example, she could be misled to believe that Naruto has some connection to a great evil that she is against.