Madara Uchiha was the mastermind behind the circumstances of Rin's death right down to Obito's apparent "accident"
If my hypothesis is correct that Madara was controlling the Mist Village/Kirigakure during The Second Shinobi World War
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Then it is logical to assume that Tobi/Obito merely inherited Kirigakure's control from Madara after the latter died
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So prior from Obito's designation as Madara's successor, Madara Uchiha himself was controlling Kirigakure for the longest time, even before he became infirm and got stuck in his cave,
You might ask, if Madara was controlling the Mizukage and Kirigakure during the third Shinobi World War (Obito's era) then how was he able to direct the Mist Ninjas in orchestrating Rin's death?
The answer is Zetsu:
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Since the Mizukage was already under Madara's genjutsu long ago, then it's only a matter of relaying his orders to the Mizukage and the Mist Ninjas through his errand boy Zetsu, and since the Zetsus can communicate telepathically, Madara need not to travel outside his cave to orchestrate Rin's demise.
I seriously doubt that Zetsu was not acting under Madara's orders when he told Obito about Kakashi and Rin's predicament, Zetsu did not warn Obito just because he considered the boy as his friend, he warned him because Madara ordered him to:
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That is why Madara was so cocksure that Obito would return to him, because he staged the events that transpired:
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There was something done to Rin's body:
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I highly doubt Rin was a spy from the beginning, more than anything she was an involuntary "sleeper spy" a Manchurian Candidate:
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It is my suspicion that the Mist Ninjas (under Madara's orders) modified Rin's body to become a sleeper spy. It could also be some sort of bomb/poison, but I'm leaning towards the former.
And since in Naruto, accidents do not happen, I also suspect that the specific rock that crushed Obito's body when he tried to save Kakashi was caused by Madara (using a jutsu to push the rock, Zetsu pushing it,etc.)
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Madara probably anticipated Obito to intervene, causing the rock to fall on him instead of Kakashi.
My suspicion is that Madara had his eyes on Obito even before he got crushed by the rock, probably watching Obito for a long time through Zetsu.
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Madara was already setting his sights on Obito to become his successor long before the incident occur.
Now regarding the death of Rin herself, it is my hypothesis that Rin herself asked Kakashi to kill her.
Somewhat Paralleling Haku's request to Naruto to kill him:
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I believe Rin found out at a later time that something was done to her body to make her an involuntary spy of the Mist Village, she did not want to continue betraying Konoha unwillingly so she herself asked Kakashi to kill her.
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Kakashi stabs her with Chidori
Look at this panel:
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The "........." text in the bubble indicates grief and remorse, both his eyes closed at the moment of stabbing her also indicates remorse, Kakashi was fully aware that he killed Rin.
Because Rin herself asked Kakashi to kill her.
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That is why she was buried in Konoha, she was not a traitor nor a willing spy, she sacrificed her life so she would not betray Konoha by becoming an involuntary spy of the Mist, she chose death instead of betraying the Leaf.
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Madara knew that as a loyal and dutiful Shinobi, Rin would choose death over dishonor.
Obito saw all of this, he killed the Mist Ninjas
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He went back to Madara:
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And Madara gets Obito's allegiance
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(Smirk)
All according to plan.
