Maybe Naruto is put in a situation where he feels that Hiruzen's way of handling things is the best option. For instance, if there is a major crisis in the village that requires a more traditional and experienced approach, Naruto might choose to act like Hiruzen. Also, he could be influenced by the stories he has heard about Hiruzen and wants to fill his shoes in a way.
In Naruto fanfiction, 'Bonds of Blood' often explores the deep family - like relationships. It could be about the bond between Naruto and his parents, or among the members of his extended 'family' in Konoha. These bonds are usually depicted as strong forces that drive the characters' actions and development.
Perhaps in this fanfiction, Naruto has some sort of special power or transformation that he unlocks just in time to face Orochimaru. It could be related to his Nine - Tails chakra in a new and unique way. He uses this power to break through Orochimaru's defenses and save Hiruzen. Naruto might also use his knowledge of Orochimaru's weaknesses that he has learned either from Jiraiya or from his own past encounters with Orochimaru's minions.
Yes, in certain fanfictions Naruto hates Hiruzen Sarutobi. Fanfiction often takes the original characters and storylines and twists them. Maybe the writer of such a fanfiction thought that Hiruzen's actions, like not fully protecting Naruto when he was a child, could lead to Naruto having negative feelings towards him in this alternate take on the story.
In the fanfiction, Hiruzen might raise Naruto properly by providing a stable living environment. He could ensure Naruto has a decent place to live, enough food to eat, and warm clothes. This basic care is fundamental for Naruto's growth.
One interesting story could be about Hiruzen mentoring Naruto in some unique jutsu that was long forgotten. This would show a different side of their relationship, focusing on the transfer of knowledge from the old to the new generation.
Perhaps Hiruzen is being influenced by some dark force or a hidden group within the village. They convince him that Naruto, with the Nine - Tails inside him, is too dangerous to be left unchecked. So, Hiruzen, in a moment of weakness, decides to betray Naruto. He could secretly arrange for Naruto to be sent on dangerous missions alone with the intention of getting him killed. Naruto, being oblivious to this at first, later realizes what's going on and has to fight not only against external enemies but also against the betrayal of his own village leader.
In this fanfiction, perhaps Hiruzen realizes Naruto's potential much earlier and decides to train him privately. He might take Naruto on secret missions to build his experience and confidence. This would not only strengthen Naruto's skills but also their bond, which could have a huge impact on the village's future.
In a fanfiction where Naruto is trained by Hiruzen, Naruto would likely have a more traditional and disciplined training. Hiruzen could instill in Naruto the values and ethics of being a true shinobi from the very beginning. Since Hiruzen has seen many great shinobi in his time, he could mold Naruto into a shinobi similar to those of the past. This might mean Naruto would be more proficient in basic ninja skills like stealth and intelligence gathering. Also, his chakra control could be improved more rapidly under Hiruzen's guidance, which would enhance his overall combat capabilities.
The plot might involve Naruto learning about the history and politics of the village from Hiruzen. Since Hiruzen has a wealth of knowledge and experience, he could be guiding Naruto on how to be a great leader in the future, not just a powerful ninja.