Earth grudge fear allows him to separate his body parts, but those parts are still "attached" via the grey tendrils. The gray tendrils are as much his body as the muscles and tendons and such of a non-earth grudgy person.
How do you not know how Shadow Possession works? If you knew only one thing it should be how shadow possession works. You are agonizing. Just stretching his arm wouldn't free him from Shadow Possession and that isn't what he does. He completely detaches his arm but places one of his extra hearts inside of it so it can move on its own.
http://naruto.wikia.com/wiki/Earth_Grudge_Fear"He could detach parts of his limbs and then manipulate them, to the point that he could even send them underground and then reattach them when he was done with them."
When did anyone ever say the chakra system would remain in a severed arm? If anything it flows out from the heart and if say, your arm got cut off, it wouldn't be able to flow there anymore.
And what is your counter-argument for the Kaguya thing? Oh it must just work anyway. They must be able to use their destroyed chakra network to heal their chakra network because the only other option is that Bocchiere is right and that can't possibly be the case. Eric you have no idea what you're talking about.
This is SL for you. I come in with a logical argument to try and discuss something and people just respond with, "I'm just going to baselessly assume there is some unknowable thing going on behind the scenes that makes your argument incorrect." Great. Good job.