I doubt Hyuuga are related to the Sage at all. And there are probably hundreds of Dojutsu ( Several in "Non canon" series/movies ).
These are blood line traits, all the Sage did was show them how to use chakra ect. He was the reason those people could activate their blood line traits, it's stupidity to assume that everyone is related to the sage.
The Sharingan was probably in his mother's blood. His father would have had the "Senju" DNA ( Who knows what it was called then ) and him being the collaboration of their DNA + The first "Natural" Chakra user simply released his Kekkai genkai.
When his children were both, one of them inherited the "Sharingan" and the other the "Senju" powers ( Things always skip a generation, right ? ), but a Senju/Uchiha Child has probably never been birthed since then.
Actually, it's stupidity to assume what you just said right here. All chakra came from one source, the tree. Kaguya was the first human to take it, and her son the Sage ended up getting it next. In order for everyone else to have used chakra, either they had to take some from the tree also or the sage had to give them chakra. This idea you have that everyone just simply had chakra in them goes against the manga.
So yes, I believe everyone is connected to the sage not via bloodlines, but via chakra. And if you consider that it was that chakra that gave him the Rinnegan, you can see it's very possible the broken down bloodline also lead to broken down chakra which resulted in other occular powers.
Chakra all came from one source, so everyone in the world is connected because they are using chakra that originated from the same source. That's a fact. Kaguya was the first to get and use it, and her son inherited it next. So yes if you consider the chakra, rather than the bloodline, everyone is connected to the sage if he was the one to distribute it to others. But make no mistake, someone had to give people the chakra. It wasn't just in them