One possible interesting plot point could be Harry introducing new magical spells or techniques that he discovers from ancient texts. This could change the way magic is practiced in the wizarding world.
One possible reason could be the trauma he has faced. Harry has endured a great deal of pain, from the loss of his parents to the constant battles and threats in the wizarding world. He might want to escape all that and find peace away from it.
In a fanfic, Hermione might initially be in denial. She would keep expecting Harry to return any day. She would go to their old haunts in Hogwarts, like the library or the Gryffindor common room, and just sit there, reminiscing. As time passes, she realizes she has to move on. She starts teaching at Hogwarts and imparts her knowledge and the values Harry also held dear to the new generation of students.
It could be that in the fanfic, Harry Potter discovers some deep - seated secrets about the wizarding world's power structure. Feeling betrayed, he decides to take extreme actions to expose and destroy it. For example, if he finds out that the Ministry of Magic has been using unethical means to control the wizarding population, he might use his influence and magic to bring it all crashing down. He could start by revealing the truth to the public, which could lead to chaos and the destruction of the established order in the wizarding world.
A 'Harry Potter World Traveler' fanfic can expand the wizarding world in multiple ways. Firstly, it can introduce new magical locations. Instead of just Hogwarts and Diagon Alley, we get to see enchanted cities in South America or hidden islands in the Pacific. Secondly, it can explore different forms of magic. There could be a type of magic based on music in some far - off land. And thirdly, it can bring in new characters from various parts of the world, each with their own backstories and motives.
Perhaps he feels disillusioned with the wizarding society. There could be issues like corruption or inequality that he can't bear anymore, so he decides to leave. For example, if the Ministry of Magic starts to misuse its power, Harry might not want to be part of such a world.
The wizarding world would be in shock. Harry is seen as their last hope, so his surrender would make many lose faith. The Order of the Phoenix might go into a state of chaos, trying to figure out how to rescue Harry and continue the fight against Voldemort without him.
It could be about Harry Potter achieving ultimate power and ruling the wizarding world. Maybe he overcomes new and even more dangerous dark forces and in the process, he becomes the king. He could make new laws and bring about great changes to Hogwarts and the entire wizarding community.
Obviously, Harry Potter himself. Hermione and Ron would most likely be there too, as his closest friends. Dumbledore might still play a role, perhaps as an advisor to Harry in his new position. And there could be new characters, like wizards and witches from other parts of the world who come to see the new king.