Let's looks at this from the perspective of another example. Admittedly future example, but maybe Kirk and Enterprise's Nil and A set a precedent.
The most obvious angle is the D and E. Somewhere (both Memory's Alpha and Beta back this up) state that the E was being built to be another ship. Then the D is destroyed and it's renamed the Enterprise and the D's Command Crew is moved across, echoing the Nil-A transition. Clearly Starfleet like keeping Officers together if they've proven themselves.
Very early on in this thread someone pointed out that the Sovereign is a smaller class, isn't also a better war vessel though? The Federation is in the middle/at the beginning of the Dominion War, the Borg threat has been looming, it makes more sense that the flagship suit that more. Getting the civilians off of a ship that is clearly going into battle is probably a good idea.
Back to the 2280s. We know that Starfleet was experimenting with Transwarp and a new class with the Excelsior, a ship that when Sulu was granted his commission he was given command of, after the Transwarp drive was deemed a failure. Seems to me that Starfleet were seriously considering a new type of ship that would outstrip the Constitution class in every way. But they had to keep their fleet up and running until they knew where they are going with this new class. They wouldn't build new Constitutions, they're expecting to build an entire new fleet when the Excelsior proves a success. So they keep the existing ones running through refits. Of course given the timescale, The A is a ship that has already gone through refit and probably about to be launched when Kirk and crew save the Earth from the Whale probe.
Kirk and crew have broken a whole lot of rules to save Spock. If that's all they had done I'd imagine the entire command crew would be up against a court martial. Given the service records it would probably be lenient. But they just saved the Earth so we'll go super easy on them, either PR, thanks or both can be claimed.
Sulu's comment about “Maybe we'll get the Excelsior” shows that rumours are that it's about to go into the field. But they can't give a crew (no matter how decorated) that stole a starship to save one man, the pride of the fleet. So they give them another Constitution with the name Enterprise. Not only will this be a huge PR spin for the public, the crew get what they want. Kirk also wouldn't see a non-Constitution class with the name Enterprise as HIS ship, he'd recognise it more as a gesture.