Even if that were the case, there’s still no inherently good reason why Kirk would need to steal that particular ship when a warp-capable shuttle would have worked just as well or even better.
Adding to the idea, how cool would it have been if Admiral Kirk got to steal the original ship because he knew its prefix code.
I had a simmilar thought, in this scenario, the Original Enterprise is the only other ship Kirk has the command codes for. My wife and I just had a great laugh imagining Scotty having to pull down velvet ropes and transparent aluminum partitions in order to turn stuff on in Engineering. Also Kirk has to remove a mannequin of himself from the Captain's chair.