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.
This fanfiction could also change the relationships with other characters. For example, if Kushina hates Minato, the Third Hokage might have to step in more often to mediate between them. And the other villagers might also take sides, creating a divided village atmosphere that is quite different from the united one in the original story.
Well, it would definitely strain Naruto's relationship with Sasuke. Since Sasuke is an Uchiha, Naruto's hatred towards the clan would put a huge rift between them. They might not be able to have the same kind of bond as in the original story.
If Kushina was alive in a fanfiction, Naruto's relationship with his father, Minato, would be different. They could all be a proper family, and Naruto would grow up with both his parents. His relationship with his friends might also change as he would have a more stable home life. For example, he might be more confident around Sakura and Sasuke, and his bond with them could be less about seeking acceptance due to his lack of family.
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.
Naruto would become more confident in his abilities. With Kushina's training, he would have a better understanding of his Uzumaki heritage and the power that comes with it. This new - found confidence would show in his battles and his interactions with others.
One possible reason could be a misunderstanding. Maybe some false information was spread about Kushina that made Naruto believe she was a bad person. For example, if there were rumors that she abandoned him for some untrue reason, it might lead to his hatred.
For Naruto and his friends, his new connection to the Dragon Sage power from Kushina could make some of his friends a bit jealous at first. However, as they see how he uses it for the good of the village, they would come to accept it. His relationship with Sasuke could also be affected. Sasuke might be intrigued by this new power and see it as a way to also become stronger, which could lead to some friendly competition between them. With Kushina, Naruto would share a deeper emotional connection as they explore their shared power heritage together.