It might be a matter of adventure. Harry has had so many adventures in the UK, but America offers a whole new continent full of unknown magical places. He's drawn to it because he wants to experience new things, meet new people, and have a different kind of adventure outside of the familiar British Isles.
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.
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.
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.
They could be on a mission to exchange magical knowledge. Hogwarts has its own set of spells and magical techniques, and so does the American wizarding world. They might go to America to learn new things and share their knowledge as well.
It could be really interesting. Maybe Harry would grow up with a different kind of magic education system in America. He might attend Ilvermorny instead of Hogwarts. His friends could be American witches and wizards with different cultural backgrounds and accents.
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.
In some fanfictions, it might be because Snape exposed a secret of Harry's that led to his departure.
Perhaps they fear that fanfiction could deviate too much from the original themes and values of the Harry Potter series. The official story has a certain moral and ethical framework, and fanfiction might distort it.
He might flee because fanfiction can sometimes distort his character. In some fan - made stories, his personality, values, or relationships could be changed in ways that are untrue to the original. For instance, if a fanfiction makes him act cruelly when he's supposed to be kind - hearted.
It could be that the Harry Potter brand wants to control the narrative more tightly. Fanfiction can sometimes take the story in directions that the official creators don't like. So, to maintain a certain image and story integrity, they might be discouraging fanfiction.