Oh, I just realized something, but I don't want to bother making an entirely new thread over it.
The reason that Obito didn't just take Edo Nagato's rinnegan was because he couldn't! The moment he chopped those suckers out, they would dissolve into papery dust and reform into Nagato's eye sockets!
Sorry if this was obvious, but it was an amazing personal revelation for me since I was wondering why he didn't just do that in the first place.