it's clearly stated in the Manga that Ashura's descendants were the Senjus only, not Senjus and Uzumakis:
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And look what it says: "even as time passed and the bloodline faded, the Brother's descendants continued fighting" and the Uzumakis were completely absent from the Uchiha vs Senju fighting.
Zetsu's statement in 606 saying Nagato is of Senju Lineage only refers to blood relations (the word Ketsuen was used in Japanese meaning "blood relative") it does not refer to descent, for there are what is called collateral blood relations/consanguinity.
That is why Kushina called themselves only distant blood relatives of the Senju, because they do not share the same apical ancestor
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