Another possibility is that there is an external force that tricks Kurama into believing that Naruto no longer needs or wants him. This could be a powerful enemy who manipulates Kurama's thoughts, showing him false visions of Naruto rejecting him. Kurama, being proud, might then choose to abandon Naruto without realizing it's all a ruse.
In some fanfiction scenarios, Kurama might abandon Naruto if he believes that Naruto is becoming too corrupted by power. Kurama has a sense of honor in a way, and if Naruto starts to use his power in a way that Kurama doesn't approve of, like becoming overly aggressive or cruel, Kurama could decide to leave. For example, if Naruto starts to use the Nine - Tails' power to dominate others rather than protect them.
Naruto would likely become more determined. He has always been a fighter, and without Kurama, he would have to rely solely on his own skills and training. He might become more stealthy and strategic, since he wouldn't have the brute power of Kurama to fall back on.
Perhaps Kakashi feels that Naruto is becoming too reliant on him and not developing his own skills independently. So, in order to force Naruto to become stronger on his own, he decides to abandon him. This is a tough - love approach. Naruto has great potential, but he needs to learn to stand on his own two feet, and Kakashi may think this is the best way to make that happen.
One possible reason could be that Sakura realizes she has different goals or values that are not compatible with being around Naruto. Maybe she discovers a new path in life that doesn't involve him, such as focusing on her own medical ninja career without Naruto's chaotic lifestyle interfering.
One possible reason could be fear. Naruto has the Nine - Tails within him, and some in Konoha might be too afraid of its power getting out of control. They may not fully trust Naruto to keep it in check.
In some fanfictions, Naruto might abandon his family due to a false prophecy that convinces him he's a danger to them. He may think that by leaving, he can protect them from harm that might come his way because of his past or enemies.
One possible reason could be some sort of dark influence or curse that makes her lose her maternal instincts. Maybe a powerful enemy targets her and in order to protect Naruto from further danger related to her, she feels forced to abandon him. Another reason might be that she is made to believe false information about Naruto, such as that he will bring great disaster to the village in the future.
One possible reason could be that Naruto discovers some dark secrets about the village leadership that go against his values. For example, if he finds out that they have been using underhanded means to maintain power.
One reason could be for shock value. Fanfiction authors like to surprise readers, and making Kurama Naruto's mother is a very unexpected twist.
One possible reason could be feeling betrayed by the village leaders. Maybe they made decisions that Naruto strongly disagreed with, like how they treated certain groups or individuals. If Naruto felt that Konoha was no longer the place that stood for justice and fairness as he once believed, he might abandon it.
One reason could be that it offers a fresh and different perspective on the Naruto character. A female version of Naruto might have unique character traits and storylines that are interesting to Kurama. For example, in fem Naruto fanfiction, there could be different relationships and power dynamics explored.