So now we're down to -A, right? That is, this wasn't the ship described either in the "Relics" novelization or in Badlands?
We can argue minutiae after that: at which point during her career would each ship have held this flagship status?
Do you want to make things complex first? :-)
Kirk noted that the Vulcan hid her own disappointment well, as she did her disapproval that Starfleet's flagship had been relegated to providing a civilian with taxi service. That's not how Captain Sulu had viewed Kirk's request for a favor, but Kirk understood how others might see it.