I think welding it to things like the arm or torso is buggy, because ball sockets move back and forth, like ropes or joints. It’s a weird issue here. What if you tried using ropes instead of ball socket constraints. Would it have a different outcome?
The chain isn’t weightless, you too would be wobbling around irl if you had a fat chain half the size of your body dangling on your torso. Make the chain weightless in the properties window, that will probably fix it.
In your first post and your reply to @xMicro_Canary you say that you have the chain Meshes Massless set to True, but in your screenshot of the Mesh Properties Massless isn’t checked off so it’s False.
You should always have anything that’s welded or connected as a tool to a Humanoid set as Massless, otherwise the Parts/Meshes mass will interfere with the way the Humanoid moves.