He could be disillusioned with the wizarding world's politics and power struggles. So, taking his money and leaving is his way of getting away from it all. He might feel that he can't make a real difference within the existing system and decides to find his own path.
Maybe he wants to keep it safe from thieves. There could be dark wizards or even some of his schoolmates who might be tempted to steal it.
Perhaps he's following a lead related to his parents. There could be some secrets about his family that are hidden in America. He goes there to uncover the truth and learn more about his family's past that might not be available in the UK.
Maybe he takes it out of a sense of duty. Harry might feel that he needs to carry on Dumbledore's legacy and that using his wand is a way to do that. He could believe that Dumbledore's wand holds the power to defeat the remaining threats to the wizarding world.
He might be holding his diaphragm because it's a part of a new magic technique he's learning. Maybe it's something taught by a mysterious new teacher at Hogwarts who believes that the diaphragm is a key element in controlling a certain type of magic energy.
Hermione could be the one. With her vast knowledge of the wizarding laws and her sharp intellect, she might dig up some old regulations that prove Harry's right to his money. She would then fight tooth and nail in the wizarding courts or against the goblins at Gringotts to get his money back.
In some fanfictions, Harry might leave because of Ginny due to a misunderstanding. For example, he could misinterpret Ginny's actions or words as a sign that she no longer loves him or wants him around. This could be something as simple as seeing her chatting happily with another guy and jumping to the wrong conclusion.
Well, it could be that the author wants to show Harry's perseverance and growth through constantly dealing with unwanted attention. It makes his character stronger.
Maybe he's constantly underestimated or not given the respect he deserves. In fanfiction, writers often explore the idea of Harry being pushed around by authority figures or jealous peers, which could lead to him leaving.
Maybe for the sake of creating drama. Fanfic writers like to add twists to the original story. Blocking Harry's magic can create a lot of tension and make readers wonder how he'll get out of the situation.
One reason could be for educational purposes. The alley is full of unique magical items and people, so the teacher might want Harry to get some hands - on experience with the real - world applications of magic.