In Naruto, the students of kakashi were taught by the three sannins ?

yes, eventually Naruto, Sasuke, and Sakura will become the next generation of Sannins which is what they probably wanted. All of them now have the ability with the a similar conflict between them. They wanted a younger version of the three to keep a moving circular story going. as one generation falls another will rise and there will always be those at the top.

I think the writer did that also, but if it were in real life then I would doubt that would ever happen. Go Team Kakashi!!!