It's not that strange. Naruto is a curious person. Since Boruto is his son, he would be curious about how others imagine Boruto's story in fanfiction. He might even find some new perspectives on his son's character and the ninja world that Boruto is growing up in. Also, Naruto has always been open - minded, so the idea of exploring fan - created stories about his family wouldn't be too out of the ordinary for him.
He could be curious about how other fans perceive the events in Naruto Shippuden. Fanfictions can offer unique interpretations and perspectives that might give him new insights.
Maybe it's the high expectations that Naruto has for Boruto. Naruto is a hero, and he might unknowingly put a lot of pressure on Boruto to follow in his footsteps. Boruto could be scared of not being able to meet those expectations.
They could start by having an honest conversation. Naruto could sit down with Boruto and just listen to his fears without judgment. This would be a first step in building a better relationship.
I don't think there would be a lot of fanfiction like that. Hinata has always been loyal to Naruto. In the Naruto universe, their relationship is based on a lot of development and mutual understanding. So it's hard to imagine a scenario where Hinata would cheat on Naruto with Boruto in a legitimate fanfiction.
In the Boruto manga, Naruto unfortunately passes away. This event has a significant impact on the story and the characters. It occurs as a result of a powerful and unexpected enemy attack.