This logic makes no sense.
You assume that the Uzumaki Clan didn't have members that allied with the Senju Clan AND TAKE ON THEIR SURNAME. Especially through marriages, the two clans would have easily had members that blended in and took on the Senju Sirname. We also see in the manga below that...
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.... The Uchiha clan members joined forces with Senju Clan members, before the creation of Konoha. I'm sure its "possible" they also assumed the Sirname "Senju" during their surrender/alliance before the actual Leaf Village was created. Whether they did or not, it obviously shows that the Senju Clan had members of other clans become a part of them... this is what I assume happened with the Uzumaki Clan. Either the Sage himself was a Uzumaki (which automatically makes them "close" blood relatives) or the Uzumaki Clan is directly descended from the Senju, not just by marriage.