The Akatsuki could also feel a sense of sadness or betrayal in this 'neglected' scenario. For example, some members might have had a certain respect or even a strange kind of hope for Naruto. So, when they are neglected, it shatters that idea they had. They might then become more withdrawn and brood over this new situation, which could lead to changes in their group dynamics and individual characters.
It could be because fans like to explore different relationship dynamics. The idea of Naruto, who is usually seen as a hero, having this flaw of neglecting someone like Fem Sasuke adds an interesting layer to their relationship.
Perhaps the Akatsuki's neglect is a part of a much larger plan by Naruto himself. He might be trying to lull the Akatsuki into a false sense of security while he's actually gathering information or power in the background. Then, when the Akatsuki least expects it, Naruto reveals his true intentions and uses the situation to his advantage, which would be a huge plot twist for the readers who thought Naruto was just being mean or indifferent.
In this kind of fanfiction, Naruto might be neglected by his village. He could be feeling alone and unappreciated. So, he decides to join the Akatsuki. Maybe he sees them as a group that could accept him for who he is, regardless of his past in Konoha. He might start to learn their ways and use his skills for their missions.
One great Naruto and Fem Akatsuki fanfiction is 'The Akatsuki's New Member'. In this story, a female character joins the Akatsuki and has some really interesting interactions with Naruto. It shows a different side of the Akatsuki dynamic.
It could be the Hidden Leaf Village itself. Since Naruto was neglected there, the village's leadership or some of the villagers who shunned him could be seen as antagonists. They pushed Naruto towards the Akatsuki by not giving him the support he needed.
Naruto's ideals of friendship and peace would likely clash with the Akatsuki's goals. His presence could make some members question their motives. For example, he might try to convince them that there are better ways to achieve power than through collecting tailed beasts. His positive attitude could be infectious and start to change the overall atmosphere of the Akatsuki.
There's a fanfiction called 'Silent Longing'. In this story, Fem Sasuke has feelings for Naruto but he is too busy chasing after power and recognition in the ninja world to pay attention to her. It shows how she copes with this neglect and her own growth in the process. Another one could be 'Abandoned Promises' which delves into how Fem Sasuke's expectations from Naruto are not met as he neglects her in their relationship.
Well, it's the contrast. Naruto is all about hope and friendship, while the Akatsuki is this mysterious and somewhat dark organization. When you add a female element to the Akatsuki in fanfiction, it creates new and exciting storylines. For example, it can explore different types of relationships between Naruto and the female Akatsuki members that you don't see in the original series.
Their relationship could start with mistrust. Naruto, having been taught that the Akatsuki are bad, would be wary of the female member. But as she shows him kindness in the face of his neglect, he might start to warm up to her. For example, she could bring him food when no one else in the village does.