Perhaps he was on a quest for a rare magical ingredient only found in PT.
It could be that he received a mysterious message or prophecy that directed him to PT. There might be some great danger lurking there that only he can prevent. Maybe it's related to a powerful dark wizard from PT who has plans to expand his influence to other magical worlds, and Harry is the only one who can stop him because of his reputation and skills.
One reason could be to study rare magical plants. Australia has a diverse range of flora, and some of it might have magical properties that are not found anywhere else. So, he travels there to learn about these plants and their potential uses in potions or other magical practices.
Harry could be following a hunch. He might have found some evidence that links Snape to a solution he needs. For example, he could have discovered a note written by his father or mother that mentions Snape's ability to deal with a certain type of dark magic that is now threatening Harry. So, he takes a chance and goes to Snape, hoping against hope that Snape will be willing to help.
Well, he might encounter new magical creatures unique to PT. Maybe he'll learn some different spells or forms of magic there.
It might be related to Hermione's research. She may have read about some unique magical phenomenon in Australia that she wants to study first - hand. And Harry, being her loyal friend, goes along with her. For example, there could be a special type of plant with healing properties that only grows in certain parts of Australia.
They might go into hiding because there are still some remnants of Voldemort's followers who are seeking revenge on Harry. These followers could be more dangerous than before as they are desperate and hiding in the shadows. Ginny, being Harry's wife, is also in danger, so they choose to go into hiding to avoid any potential attacks.
They might go to the Burrow to reconcile. Fanfiction often explores the idea of enemies becoming friends. The Burrow, being a place full of warmth and family, could be a good setting for them to start over. It's a place where they can put aside their differences and get to know each other better without the pressure of Hogwarts politics.
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.
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.