I have used the same Google account on four devices and co-signed on to my wife's phone, and none of that required swapping components around. People switch phones and have multiple devices with the same account all the time and never have to open up the phone to do it. If you are having login issues, could you please elaborate more?
I know you're coming from a G4, but what are you trying to move to?
How did you get the new device? (I.e. Upgrading, bought used, repair service, etc)
When do you get the error, what is the exact wording of it? It would also be handy if you can upload a screenshot of this error.