1 Kakashi passed out because MS was activated
2 Rin was either senju or uzumaki and Kakashi didnt want Rin to suffer toture and experimentation
Plot Hole: It doens't make sense for Obito to not kill Kakashi either in the next chapter OR during the massacre of the Uchiha clan
If Rin was an Uzumaki, a reason to get her exist.
What are exceptionally good Uzumakis for?
They are perfect for becoming Jinjuriki.
Maybe the kage of the hiden mist was unstable, and a new host was needed.
Or the elite of the mist knew about Madara controlling their kage, and after killing him a new host was needed. The ANBU needed only her. And losing her was a great disgrace.
But there is a little problem.
So much important she is for Konoha, and they send a kid to
protect/kill/liquidate whatever her.
If she was a Senju, she don't need protection because they are not weak at all.
Even in death they are powerful, they are like steroids for ninja. And again why send a kid to protect a so important thing?
And for Obito...
Maybe he teleportet all the ANBU somewere else, or that's all an illusion, or Kishi is drawing half drunk/half slept.