There are multiple reasons. Firstly, the Third Hokage was highly respected in the village, and Naruto, being a part of the village, would naturally be influenced by this. Secondly, his wisdom in handling village politics and relations with other villages is something Naruto could look up to. In fanfiction, this could be explored more deeply, like Naruto learning from the Third Hokage's diplomatic skills to better protect the village in future interactions with other villages.
The Third Hokage was a symbol of stability in the village. Naruto, who grew up without parents and was sometimes an outcast, would see the Third Hokage as a source of strength and guidance. Also, the Third Hokage's knowledge of various jutsus and ninja arts could be something that Naruto admires in fanfiction, as Naruto is always eager to learn new techniques.
Perhaps it was related to Naruto's heritage. The Third Hokage might have been influenced by some hidden secrets about Naruto's parents. If there were things he didn't want to come to light, he could have betrayed Naruto to keep those secrets buried. For example, if Naruto's father had enemies within the village, the Third Hokage might have tried to keep Naruto at a distance to avoid those old grudges resurfacing.
One reason could be the overwhelming pressure. Being Hokage means shouldering the safety and well - being of the entire village. Naruto might have realized that he can't bear that kind of stress forever. Another factor could be personal sacrifice. He may have seen how being Hokage has cost others in their personal lives and he doesn't want to go through the same. Also, he might have new dreams or goals that are not compatible with the role of Hokage, like exploring the world outside the village more freely.
One possible reason could be the overwhelming stress. Being Hokage means handling a lot of political, social and security issues in the village. Naruto might just feel that he can't take it anymore and decides to quit in the fanfiction. For example, dealing with the constant threats from other villages and internal disputes within Konoha can be mentally and physically exhausting.
One possible reason could be that some elders in the village still have doubts about his past. For example, his connection with the Nine - Tails. They might fear that this could lead to instability in the village. Another reason could be political intrigue. There might be other powerful clans or individuals who want the position for themselves and manipulate things behind the scenes to deny Naruto.
One possible reason could be that Naruto has seen the toll the position takes on those who hold it. Maybe he witnessed the stress and loneliness that previous Hokages like his father had to endure. He might want a more carefree life, exploring the world and spending time with his friends and family without the heavy burden of leading the village.
Perhaps Naruto feels that the Hokage has been too strict or unfair in dealing with certain situations. Like if the Hokage didn't support Naruto's way of trying to bring peace and unity, which Naruto firmly believes in. This could lead to a growing resentment and eventually hatred.
Yes. In some fanfictions, Naruto could be depicted as having great admiration for the Third Hokage. The Third Hokage was a highly respected leader in the village. Naruto, growing up in the village, would have heard many stories about his leadership, wisdom, and sacrifice. This could lead to him being a fan in the context of fanfiction.
One reason could be that there are political forces against him. Maybe some of the elders in the village don't trust his unorthodox ways and fear he'll bring too much change. Another could be that there are other candidates who seem more suitable on the surface, perhaps someone with a more traditional shinobi background who doesn't have Naruto's wild streak.
One possible reason could be the pressure associated with the Hokage title. Naruto has always been in the spotlight and being Hokage comes with a huge burden of responsibility. Maybe in the fanfiction, he resents how it restricts his freedom to just be himself and have adventures like he used to.
In this fanfiction, Naruto could be given a special mission by an unknown entity to save the Third Hokage. This mission might push Naruto to his limits and force him to unlock hidden potential within himself. Along the way, he might face challenges that test his loyalty and friendship.