I would suspect that if the restore is not going through it would be more likely to be charge port or the logic board, there may be some sort of communication error with necessary components. I don’t suspect battery is at fault here, usually an iPad with a faulty battery will still give some signs of OS function, even if it’s just going into recovery mode. In my experience an error 4005 is a “not a good error to get” error and I would be leaning towards logic board if you can’t get it to do anything other than interact with iTunes in DFU mode.