When I say Senju dna I was referring to Ashura's chakra (hashirama's dna containing it). So you're right about that.
However the bold part, Ashura incarnates receiving Indra's chakra does not allow them to awaken the rinnegan, It's completely false.
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You're completely wrong. Black Zetsu sought for Transmigrants of
both clans to awaken the Rinnegan because each had the potential of doing so. The reason why Indra Transmigrants were more viable wasn't because they were the only ones who had the potential to do so, rather that Black Zetsu set the foundations for the Uchiha Clan to be more willing to fulfill the plan to revive Kaguya hence why he rewrote the Uchiha Tablet to convey that Infinite Tsukyomi would save their Clan:
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Hagormoo specifically left the tablet with Indra's people because he said that they were the only ones who had the potential to awaken it.
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You're wrong. He knew that Indra's Transmigrants were more likely to try to attain more power through attaining the Rinnegan, so he left the Stone Monument for them to reconsider.
Tobirama was completely unaware and he was still studying their potential.
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No, he was basing off his analysis of the Uchiha's Power through his observations.
Just because an uchiha retrieve another chakra to their own doesn't mean it cannot be within their potential. People learning to acquire senjutsu is on the same boat as uchiha's getting senju dna. They need an outside source to make it permanently there own.
Senjutsu can be attained through normal means such as training. A Transmigrant attaining the other component of Rikudou's Chakra isn't something that can naturally be done through training.