One possible reason could be extreme disillusionment with the wizarding society in Britain. Maybe he faced so much political intrigue, discrimination, or ungratefulness from the people he had saved that he just couldn't take it anymore. For example, if the Ministry of Magic continued to be corrupt and turn a blind eye to dark magic activities, it might push Harry to abandon the place.
Perhaps he felt that his presence in Britain was endangering those he loved. Voldemort's followers might still be after him and targeting his friends and family. By leaving, he could draw the danger away from them. Like when Sirius died, he might have thought that staying put would only lead to more losses.
It makes him seem more self - centered at first glance. Traditionally, Harry was seen as the savior of Britain, so leaving it to its fate goes against that heroic image. But on a deeper level, it could also show his growth into a more complex character who has his own needs and boundaries.
One reason could be to escape the constant scrutiny and expectations that come with being 'The Boy Who Lived' in Magical Britain. He might feel suffocated by the fame and the never - ending battles against the Dark Arts in his homeland. Another reason could be a quest for new magical knowledge. Magical Britain has its own set of magical teachings, but there could be other, more esoteric forms of magic in other parts of the world that he wants to explore. Also, he might be running away from a personal tragedy or a broken relationship. Maybe a falling - out with his friends or a loss of someone dear to him makes him want to start anew elsewhere.
One reason could be that the official Harry Potter canon is so extensive and detailed that it overshadows fanfiction. The original story has such a rich world, characters, and plotlines that it becomes the main focus for many fans, leaving less room for fan - created stories.
In fanfiction, Harry could flee Britain due to a sense of disillusionment with the wizarding world there. He has been through so much trauma and has seen the flaws in the system. For example, the way the Ministry handled the threat of Voldemort in the past was often inept. He might also be seeking a place where he can explore magic in a different way, free from the traditional constraints of the British wizarding community. He may want to start anew in a place where no one knows him as 'The Boy Who Lived' and can just be Harry, an ordinary wizard exploring his magical abilities.
One reason could be to escape the constant reminders of his past traumas in Britain. The place is filled with memories of Voldemort and the battles he fought. Maybe in the fanfiction, he just wants a fresh start away from all that chaos and danger.
Hermione could accompany him. She's always been by his side and would likely follow him out of loyalty and a thirst for new knowledge. Together, they could explore new magical theories in different parts of the world. Another possibility is Hagrid. He has a soft spot for Harry and might want to protect him on his journey. He could introduce Harry to different magical creatures in other countries.
One possible reason could be that he's tired of being used by the British wizarding world. He has faced so many challenges and been through a great deal of trauma, like fighting Voldemort multiple times. He might feel that it's time for others to take responsibility and that he wants to focus on his own life for a change.
Perhaps he leaves Gryffindor in fanfiction because he is influenced by a new mentor or friend who is from a different house or group. This person might convince him that he can achieve more or be more true to himself outside of Gryffindor. Also, it could be due to a prophecy or a special mission that requires him to distance himself from Gryffindor for a while to protect the house or the entire Hogwarts from a certain threat.
Perhaps in the fanfiction, Harry Potter runs away from Britain because he is disillusioned with the way the Ministry of Magic operates. There could have been a major event where the Ministry fails him or betrays his trust. Also, he may be seeking a place where he can explore his magic without the constraints and expectations that come with being 'The Boy Who Lived' in Britain.
One reason could be that he discovers a new threat and decides he can't stay with the Dursleys any longer for their safety. Another might be that he simply can't bear their mistreatment anymore and finds an opportunity to escape.