I cannot say you are wrong because you built up a certain logic, and I like the way it came out. Also, while I don't tie the Uzumaki to SO6P directly as his child, I think there is a strong connection between the two.
However, I found a few not-so-well tighten screws in your logic:
1. Uchiha and Senju from Elder son cannot be possible.
This defeats totally your theory, and you might not accept it. However, many people have been misinterpreting Kabuto's statement about "What lies beyond the sharingan" as Rinnegan being the final state of Sharingan. As far as I am concerned, I disagree with this interpretation.
No matter how you reasons, both Rinnegan and Sharingan are the powers or chakra from Juubi/Shinju in its perfect form. They both coexist within its eye, which means none of them is a mutation of another. They are two separate type of chakras. Thus, what Kabuto did was to reconstitute the Rinnegan chakra by adding an Uchiha's part of that chakra to the Senju part of the chakra. Specifically, I am saying that what Kabuto mixed was not EMS +Mokuton. Rather, it was another cahkra that Madara had that was mixed with Hashirama's. The reason is that after unlocking his Rinnegan, Madara still have his EMS unchanged, which means that EMS never mutates to Rinnegan.
2. Misinterpretation of SO6P statements to the Bijus!
Here, you might be misinterpreting SO6P. Note that the Bijus together can be considered as Juubi. Also, by hosting the Juubi, the SO6P might have noticed 9 different spirits composing the beast. Thus, he divided it into those nine spirits. While you can disagree, it is logical to believe that SO6P did not create the 9 bijus randomly. Rather, he created them from a knowledge that they form a perfect division of the Juubi.
Therefore, if juubi had already these 9 spirits in him just as the current Juubi is 9 bijus in one, we could understand that SO6P himself could say, "unlike when you were inside me ..." Thus, I think the person who talked to the Bijus was SO6P himself.
Remark: If you look at the body of Juubi, he had 9 eyes (Gedo Mazou). Gedo Mazou is the actual Juubi SO6P fought, and when he got his spirits via Akatsuki, he decided to go towards his original form - the Shinju. The 9 eyes suggest that he had 9 spirits inside him. In facts, Kyuubi and Gyuki knows Shinju and the Juubi being progenitor. How do they know if they never seen it? They had to exist withing Juubi. I bet there is story waiting ahead.
3. Nagato's rejuvenation.
Nagato did not rejuvenate because he is Uzumaki. If it was that, he would have rejuvenated when he was alive. Rather, Nagato rejuvenated when he sucked Naruto's KM chakra along with KB's. We know that Naruto's KM chakra was all life force. Thus life came back to Nagato's body. Similarly, there is no Uzumaki DNA in MAdara's EDo. Rather, he has full Hashirama's clone being his vessel. Thus, knowing that Hashirama was all life force, there is no need to bring the Uzumaki.
How many Uzumaki could regenerate without hand seal?
--- My Opinion ---
I have said, and I maintain that the Uzumaki have a lower version of the combined Uchiha and Senju. They have a large life force, strong body, and strong spirit (Will) and chakra. As you listed, all the Uzumaki shown so far have something special. In my opinion, their ancestor has the same parent as SO6P - that is they are also Kaguya's children or her husband children. While Madara said that Rikudo was Kaguya's son, he never said that Kaguya did not have another son or a daughter. Adam and Eve had more than one child, and it appears that Kishi is not far from them - thought the story are not the same.