Sage isn't Uzumaki. He was the byproduct of *****, **** juice, and a really powerful fruit. Then he used Creation of all things to create two children with half of the Juubis chakra each.
Yin to the Uchiha ancestor. Yang to the Senju ancestor. They each had children. The Uchiha ancestors children gained byakugan at birth. Then they're children split and one half decided to keep it in the family, Hyuugas, and the other half gained regular eyes but had the potential for Sharingan.
The Senju ancestors children got traits of the Yang chskra at birth before they split. Long life force, Uzumaki, and strong bodies and jutsu mastery, senju.
The sages descendants became Uzumaki, Hyuuga, Senju, and Uchiha but he was never any of those bloodlines