Another reason could be related to Kakashi's past actions. If there are some secrets or things Kakashi did in the past that come to light and Naruto doesn't like them, he could snap at Kakashi. Naruto has a strong sense of justice and loyalty, so if he feels betrayed or misled by Kakashi, his reaction could be to snap.
One aspect of Naruto and Kakashi family fanfiction could be the idea of Kakashi mentoring Naruto not just as a sensei but as a family member. This could involve Kakashi helping Naruto understand the importance of family values, which Naruto lacks growing up. In fanfiction, we could see Kakashi inviting Naruto to family - like gatherings with Team 7, making Naruto feel more included and part of a family unit.
If Kakashi adopted Naruto, it might change Naruto's path of development. He could receive earlier and more specialized training, and also have a stronger sense of family. This could have a significant impact on his personality and goals.
Their relationship is like a mentor - mentee bond. Kakashi has trained Naruto, and when he sees Naruto achieve great things, it can be overwhelming. For example, when Naruto masters a new jutsu that Kakashi thought was difficult for him, Kakashi might cry out of joy for Naruto's accomplishment.
It could become strained. Naruto snapping at Kakashi would create an initial rift between them. Kakashi might be taken aback by Naruto's outburst and Naruto, after snapping, might feel a bit guilty but also stubborn about his feelings, which would make the relationship tense for a while.
Naruto might be the better singer in terms of energy and enthusiasm. His loud and boisterous nature could translate into a powerful and engaging singing voice. He'd probably belt out songs with great passion, getting the audience pumped up.
Sasuke could be a great addition. He could be a rival to Naruto even at a young age, and Kakashi would have to deal with their constant bickering. His brooding nature would add an interesting dynamic to the story.
Their relationship would likely become very close. Naruto would probably start to see Kakashi as a father figure. Kakashi, in turn, would become more protective of Naruto.
Well, Kakashi could train Naruto in combat skills specific to ANBU missions. This would include more lethal and precise attacks. He might also teach Naruto how to use different types of weapons that are common in ANBU operations. Maybe Kakashi would set up mock battles where Naruto has to face multiple opponents at once, just like in real ANBU scenarios.
One possible reason could be that in the fanfiction, Kakashi might have made a decision during a mission that Naruto strongly disagreed with. For example, Kakashi could have chosen to sacrifice something important to Naruto in order to achieve the overall mission objective.