The dynamic between the characters would be completely different. Without Naruto, Sasuke would lack his main rival within the team. Their growth as characters might be stunted as they wouldn't have that constant push from Naruto. Also, Sakura would be missing an important friend and source of motivation.
One possible reason could be a sense of being undervalued. Maybe he felt that his teammates didn't fully appreciate his efforts during the Wave mission. For example, if Sasuke got most of the credit, Naruto might feel pushed aside.
He might face disappointment from his friends and mentors. His friends who were also participating in the exam might feel let down, and his mentors, like Kakashi, could be concerned about his decision and how it will affect his future as a ninja.
Naruto faces a whole new set of challenges. Firstly, the ANBU has a strict code of conduct which he needs to adapt to. The missions are often solo or with very small teams, so he has to rely more on himself rather than his comrades like in Team 7. Also, the enemies he faces in ANBU are often more powerful and cunning. He has to keep his identity a secret, which is a big change from his more open - natured role in Team 7.
Well, his relationship would become quite strained. Sasuke might feel betrayed as Naruto was an important part of their dynamic. Sakura would be confused and sad. She had come to rely on Naruto in many ways. Kakashi, as the sensei, would probably feel like he failed Naruto in some way. Naruto's departure would leave a big gap in the team, and there would be a lot of unspoken emotions.
It might be related to a connection with some powerful entity in the Wave arc. For example, if there's a secret shinobi from the past whose power is tied to the Rinnegan and Naruto somehow forms a bond with that power source, it could lead to the awakening. Also, the intense emotions and challenges he faces in the Wave arc might be a catalyst, pushing his chakra to a new level and awakening the Rinnegan.
As the daimyo, Naruto would face numerous difficulties. For starters, the cultural differences within the Wave Country itself could pose a problem. There might be different ethnic groups or regions with their own traditions that conflict with Naruto's plans for unity. Moreover, he has to deal with the after - effects of any previous wars or conflicts in the Wave Country. Rebuilding damaged areas and rehabilitating the affected people would require a lot of resources and careful planning. Additionally, he has to balance his ninja duties (if he still has them) with his new role as daimyo, which could be a real headache.
Maybe he realizes the exam is too dangerous or he discovers some unfairness in it. In the original Naruto story, Naruto has faced many difficulties and moral dilemmas during the Chunin Exam. So in the fanfiction, similar situations might push him to quit.
The storyline would be completely different. Naruto's character development would take a new turn. He would no longer have the instant boost of power from the Nine - Tails. This might make him more vulnerable, but also force him to rely more on his own skills and training. His interactions with the other Jinchuuriki would also change. They might view him as a traitor to their kind or as a pioneer for a new way of living without the tailed beasts.
One possible plot twist could be that Naruto discovers a new power source that makes being a Jinchuuriki seem obsolete. Maybe he finds an ancient artifact or a new form of chakra that is more powerful and doesn't have the drawbacks of being a Jinchuuriki. Another could be that he has a moral dilemma about the nature of the Jinchuuriki system and decides to quit in protest. For example, if he realizes that the Jinchuuriki are being used as tools more than protectors.