I can counter with Obito plainly stating Kakashi himself awoke it.
Not really, If you go back and read that chapter then you would first see the distance Obito is at from Kakashi here
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and second, you will see both of them are looking down as they got the MS
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. So how would Obito know Kakashi got the MS? Obito was being overwhelmed by what he just saw as you can see, the darkness that Tobirama talked about is showed by Obito as he killed the mist ninjas
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. Kakashi fainted while Obito's Hatred drove him, so Tobirama's words go along with what happened and again, how would Obito know that Kakashi got MS when he did? I already showed that neither one was even look up and that Obito was only seeing Kakashi's side from a good amount of distance away when he was looking up.