Its just not in the story's nature to KILL the entire crew of the Yorktown.
Well, Nimoy once claimed that nobody died in ST4:TVH, this being part of the charm of the movie.
He was simply wrong. Millions must have perished in the extreme conditions shown. It's just something inherent in the setup, a necessary corner the writers wrote themselves into.
Something tells me that Starfleet would want to honor their memories. I don't think renaming their ship and giving it to Kirk would be right.
If so, Starfleet would be radically different from real military services. Yet, Starfleet in TOS was the archetype of military services - sometimes a caricature, even.
The 600 dead might get a memorial somewhere; possibly a fancium-forged plaque on the wall of some mess hall. There might be a commemorative coin or something. But if there also was a spare starship lying around unused, it would be put to use without much ceremony. And while Starfleet would not be interested in having another Enterprise
, the PR folks of the UFP would be telling them to have one so that people would be a bit happier about the more relevant million or so dead - so this spare starship would be a pretty nifty solution.