You're missing the point. Sharingan, at least in my opinion, grants the user the ability of insight, while the Byakugan grants the user the ability of visiblity. Let me elaborate. A Sharingan user can, for example, see and understand what technique his opponent will use, and therefore also understand that he either needs to move out of the way, or counter it. The theory that it grants automatic reactions is utter bullshit. It is a tool, not a controlling device. The Sharingan isn't doing the fighting. On the other hand, a Byakugan user can see around himself (referring to Neji) in pretty much a 360 degree circle. He can see, he can predict, but he can't see what technique person x is going to use. He does not know what he needs to do to counter it, or how for that matter. Sharingan can see some and understand, Byakugan can see all, but not understand

Also, the powers of these eye techniques depends a lot on the user. Experience and strength must be huge factors in how well they help the user. Sasuke still got his ass whopped by Lee with the Sharingan. Even if you can see it doesn't mean you can do anything about it. It's like tripping and falling into the road infront of a comming car; you know it is going to hit you, but you can't do anything.

While technically anyone could learn the Chidori and Rasengan, they may not be able to use it. The fact that Kakashi and Sasuke use the Chidori at high running speeds does not mean it can't be used in other ways. And while neither Naruto or Jiraiya used the Rasengan at high running speeds, it does not mean it can't be used that way. Just because one thing is one way doesn't mean there can't be another way.