Should we be reading the manga?

[Abdula]

I'm pretty sure every Hyuuga has the Byakugan. That's why the Branch and Main families exist in the first place, isn't it?

Yes, every single Hyuuga is born with the Byakugan.

But alas, Kakashi says "The ability to control two elements simultaneously, and create a new element to manipulate, that's what's called a kekkei genkai."

Even though Kakashi said the ability to control two elements simultaneously and create a new element is called a Kekkai Genkai, that does not mean that every Kekkai Genkai is a mixture of two elements. We've seen some like Haku's or the First's that are a mixture of two elements and then there are others like the Byakugan or Kimimaro's that don't appear to use any elements whatsoever.

It could be like the Sharingan, in that most Uchiha are Fire Element guys, and require Lightning Element to use the eye.

No one ever said the Uchiha's need lightning element to use the sharingan. As far as has been shown Sasuke is the only one who used lightning element and he unlocked the sharingan long before he ever learned of his affinity for lightning. Plus although it was said that the Uchihas have an affinity for fire techniques we have no idea if that is in anyway related to the sharingan.