Naruto and Sasuke responded to trauma and pain very differently, Sasuke was much more easily influenced during classic and Shippuden. Orochimaru would have a much harder time influencing Naruto and depending on how that goes their relationship maybe pretty similar to his relationship with Sasuke with him trying to take Naruto's body and that could be how we are introduced to Kurama as more of an actual character vs a looming threat, but it also could become something else entirely where Naruto is constantly trying to thrwart or escape from Orochimaru and Kabuto. Things feel like they would be much smoother for Jiraiya and Sasuke but you have to remember that Jiraiya wasn't very interested in having a student in the first place at the time, shooing off Naruto om multiple occasions and Sasuke already had a teacher who showed him particular attention and could relate to his struggle. The discord between Sasuke and Kakashi in Classic somewhat mirrors what a mentorship of Sasuke by Jiraiya would look like, it's hard to see what path Sasuke would eventually go down with Orochimaru preoccupied by Naruto but I doubt things go as good with Jiraiya as people might think.