Another possibility is that Harry has a strong connection to the official franchise and doesn't want to get involved with something that might not be officially endorsed. Fanfiction exists in a sort of gray area, and he may not want to support or engage with it. Additionally, he could be worried about the potential legal issues associated with fan - created works, even if they are often overlooked. He just wants to keep his love for the story within the boundaries of the original and legal framework.
He might not like the idea of different people having so many different takes on the story. Fanfiction can be extremely diverse, with various versions of characters and events. Harry may find this chaos and lack of a single, unified vision off - putting. Also, some fanfiction can be quite outlandish in its concepts, and he may prefer the more straightforward and established narrative of the original work.