Well, it could be that Harry, at that moment, saw Draco not as his rival but as a fellow student. In the wizarding world, Quidditch can be a very dangerous sport. Harry might have thought about how his friends would feel if he were in Draco's position. Also, Harry has a heroic streak. He might have instinctively jumped into action without really thinking about who it was he was saving. His first impulse could have been just to stop a potential disaster from happening on the pitch.
In a romance fanfic scenario, Draco could save Harry from Dumbledore if there is some sort of misunderstanding. Dumbledore might be acting on information that Draco knows is false. Since Draco has fallen for Harry, he steps in to prevent harm. For example, Dumbledore could be about to send Harry on a dangerous mission that Draco knows is a trap. Draco's love for Harry makes him take a stand against Dumbledore.
A fanfic could focus on Harry's post - Hogwarts life as a Quidditch star. He could be traveling the world, playing for a professional team. Maybe he's using his fame to promote unity among different wizarding communities through the sport. He might also be mentoring young players, passing on his knowledge and love for Quidditch.
Dumbledore, as Harry's guardian in the fanfiction, would probably use his influence in the wizarding world. He could make sure that Harry is taught by the best teachers at Hogwarts. He'd also be there to offer advice and guidance, especially when Harry faces challenges like dealing with Voldemort or other dark forces. For example, he might teach Harry some advanced defensive magic privately to better protect himself.
Well, in some fanfictions, Harry might attack Dumbledore if he discovers that Dumbledore has been using him as a pawn in a bigger game all along. This feeling of betrayal could drive him to lash out. Also, if there is a prophecy that he misinterprets to mean Dumbledore is a threat to his future or the future of his friends, that could be a motive.
Snape can be a prominent character. Given his complex relationship with Dumbledore and Harry, his perspective could add a lot to the story. He might have some insights into Dumbledore's actions that Harry doesn't know about, and his reactions to Harry's blaming could vary from being scornful to having a moment of understanding. Luna Lovegood could also be a part of it. Her unique view of the world might make her offer some different viewpoints on the situation between Harry and Dumbledore.
Well, their love for Quidditch gives them a common ground. They understand each other's passion for the sport. When they're on the pitch, they can communicate without words, just through their actions and teamwork. It's like a special language only they share. This mutual understanding deepens their bond.
Dumbledore might give Harry advanced spellcasting training. He could teach Harry some of the more complex and ancient spells that are not commonly taught at Hogwarts.
Hermione could be a main character. She might use her vast knowledge to help Harry and Dumbledore in their adventures. For example, she could research ancient spells or potions that are crucial for their mission.
Perhaps there are safety concerns for the students that Dumbledore is not addressing adequately. Harry, having a strong sense of responsibility for his fellow students, may feel that the only way to ensure their safety is to remove Dumbledore from his position. He might believe that Dumbledore's actions or inactions are putting the students at risk, and thus he takes matters into his own hands.