One reason could be to explore Sakura's character development outside of the traditional Team 7 dynamic. By having her separated from the team, it allows the writer to focus on her individual strengths and weaknesses.
They're popular because Sakura is a complex character. When Team 7 leaves, it gives a chance to explore different aspects of her personality. For example, her strength and independence can be highlighted in ways that might not be as obvious when she's in the team.
There might be a fanfiction where Sakura and the members of Team 7 go through a time - warp and end up in the past. They then have to figure out how to get back to their own time while not changing the course of history too much. This could involve a lot of character development as they face challenges they never expected.
One popular 'Team 7 Leaves Sakura' fanfiction might be one where Sakura goes on a solo mission and has to prove herself without the immediate support of Team 7. It could explore her growth as a ninja in that context.
One good one could be 'Sakura's New Path'. In this fanfic, after Team 7 disbands, Sakura decides to focus on her medical skills and ends up traveling the world to learn from different healers. It's a great exploration of her character growth outside of the team.
Another reaction could be that Sakura tries to follow Sasuke. She believes that she can change his mind or that she can be of help to him. She might leave her own responsibilities behind for a while to chase after him, thinking that their bond is strong enough to overcome whatever is making Sasuke leave.
She may feel angry at first. Sakura is a strong - willed character. She might think that Naruto has made this decision without considering her feelings. In the fanfic, she could go on a rant about how he should have talked to her before making such a big decision.
There's a fanfiction called 'Sakura's Battlefront' which is full of action. In it, Sakura and Team 7 are thrust into a war - torn land where they have to fight off waves of powerful enemies. Each member of the team gets to show off their combat skills in intense battles.
Sasuke might initially be indifferent on the surface, but deep down he would feel a sense of loss. He has always had a rivalry with Naruto, and Naruto's departure would mean an end to that dynamic. Sakura would be devastated. She has grown to care about Naruto as a friend and teammate, and his leaving would leave a big hole in the team.
Obviously, Sakura herself is the main character. Naruto could also be in it. Maybe he tries to convince Sakura to stay or he could be the one to help her prepare for her journey. Sasuke might be another character. He could have some influence on Sakura's decision to leave, either by his own actions or just his presence in the village.