Personally, I don't care that much whether the sage's clan was Uzumaki but here's a possible explanation if he's an Uzumaki.
We know this:
1) Sage has two sons, their descendants are the Uchiha and the Senju.
2) Sage traveled the world teaching Ninshu and explaining chakra, which then became Ninjutsu. This explains how shinobi came to be. The sage being born with chakra might refer to him having a special kind of it other than what ordinary people have because of the tree.
So in order for the Sage to be related directly to the Uzumaki, the Uzumaki clan's ancestor may have been:
1) A sibling of Kaguya Ōtsutsuki.
2) A sibling of Hagoromo Ōtsutsuki.
3) Another child of Hagoromo Ōtsutsuki that is unknown yet (perhaps because it was a female and didn't affect the rivalry between the younger and older son).
I think that the sage having other children than the older and younger sons is possible because of the existance of kekkei genkai, which I think is a sign of DNA mutation as a result of receiving something from the Shinjuu. Something that supports this is that some Bijuu, like Son Goku, have kekkei genkai so this might be a hint.