I don't see what harm it can do to leave the old GEC connected. I have done many service changes where the meter or the panel are moved.(As i am sure you have also)
I have utilized the old GEC and electrodes simply because i beleive more earth contact will achieve the 25 ohms or less ruling, and as you've stated, the possibilty of a backup.
GEC's can be installed in singular runs, so by code we could have as many seperate GEC runs as electrodes we desire.
If the original GEC has been installed correctly, the only possible parrallel path would be back to the serving utility X-former through common water or other mettalic lines in the event of a noodle faluire ( old ground, i know)
The event of a lighting strike or other power surge would, in theory, see ampacity divide among the many GEC's