As Rick Sternbach put it in the TNG TM, "If you could replicate an entire starship with the push of a button, you wouldn't need to."
Steering it back around to the original question, and looking back from TVH instead of ahead... There is less-than-reliable data as to what Kirk did with the Enterprise after TMP. We know, from Generations, that Kirk left Starfleet between TMP and TWOK. We know he came back not too long before TWOK, and seems to have become an Academy instructor, with the Enterprise in semi-retirement, repurposed as a training vessel. At that point, it was just another Constitution-class starship a bit past its prime, and he was just another Starfleet officer with the stuff to make it to Starship command, and thence to the Admiralty.
As was pointed out, as of TWOK, Chekov and Kyle were on the Reliant, Spock and Scotty were on the Enterprise in a training capacity, McCoy and Uhura seemed to be at the Academy with Kirk (a sense I get, from stuff in both TWOK as well as TUC), and Rand and Chapel were working at Starfleet HQ. Sulu was slated to be Captain of the Excelsior once she entered active service, but it's unclear what he was up to in general before Kirk roped the gang into helping out with the new class' Kobayashi Maru test and subsequent training cruise.
All that to say, Our Heroes were all getting on with their careers. But then the Reliant stumbled across Khan and everything went pear-shaped. Because of what happened there, and its aftermath, Kirk ended up disobeying direct orders, Scotty sabotaged the Excelsior, Sulu and Uhura assaulted Starfleet personnel -- and then Kirk, with the help of Scotty, Sulu, and Chekov, stole the Enterprise, Killed a Klingon renegade and his crew (however justified, it'd strain things between powers), and then deliberately destroyed the Enterprise, again with Scotty's and Chekov's assistance.
So those guys were on the outs with Starfleet at the time of the probe incident. They probably wouldn't have held McCoy responsible, due to his diminished mental state, and Spock certainly not. Saving the planet helped, yes. Note that all the charges were dismissed but the one on Kirk for disobeying orders. But that's still a black mark for all of them to work off. The Federation Council might be ready to hang medals on all of them, but Starfleet Command, and their fellow officers, would be looking on them with mistrust until things had settled. I still have @Timo's old essay on this, from rec.arts.startrek.tech, and I still largely agree with it. The Federation Council wanted Starfleet to reward Kirk with a starship command again, but he hasn't been in command of a starship in even longer than between TAS and TMP. He already retired once. He's halfway out the door. So they're not going to give him something just out of the yard.
Whether the Enterprise-A was an old, refitted Constitution or a new-built uprate, it's still old tech by this point. It's a posting for Kirk to finish out his term and for the others to serve their probation. It says something that Sulu still got the Excelsior, that Uhura still went back to the Academy and HQ, etc. Personally, I prefer it being an old ship renamed. As Timo put it, it was given to Kirk as a replica of the Enterprise he'd lost, but they couldn't paint "1701" on the hull, as that number belonged to the original. So it got the suffix to distinguish it -- but in fleet inventory, it was still on file as spaceframe NCC-1704, or whatever. With an annotation as to its changed displayed pennantry.
After Kirk thwarted the conspirators and basically single-handedly assured the peace treaty between Federation and Klingon Empire that would lead, in a few decades to the Klingons "joining the Federation" (as TNG put it), what notoriety the names "Kirk" and "Enterprise" had got squared and cubed. From his time as head of Starfleet Operations, he would have been familiar with the Great Experiment, if the Excelsior took as long to launch as the Galaxy. Maybe decommissioning all the Constitutions was an early concession. Per Torg in TSFS, Klingons regarded them as "Battlecruisers", while the Excelsiors were new enough and there had been no active engagements with the Klingons with them as there had been with the Constitutions. Maybe it was just that ship, with her age and wear and tear.
At any rate, Kirk went to work on turning the Excelsior into something less combat-oriented, in the process creating Starfleet's first "Explorer" (a term later applied to the Ambassador and Galaxy classes). Starfleet decided to capitalize on his increased fame and involvement by naming the first ship thus configured Enterprise, and carrying the registry forward again, now with the "-B". The launch of that ship was very definitely a PR event for Starfleet.
And then, as Timo put it, "like all stupid military traditions, it didn't die the death it so richly deserved".
As for the Enterprise-A being the Yorktown, it works. Especially if the ship required repairs or even a minor refit after its encounter with the probe, and also given ancillary sources have an Excelsior-class Yorktown entering service around then. I'm not wedded to the idea, but it works. In my notes I have a little thing about newer ships being considered, such as the newly-finished Levant or Ti-Ho, before Starfleet decided to go with the older ship. I tend to like to include as much as I can from all sources, massaged to fit.
