The structure of the Excelsior also changed along with it's registry number. If there were already more than one of her then why would Kirk refer to the ship as an experiment? And I have to disagree with the notion that Starfleet transferred the command crew off a ship to simply have better PR. Because that would be assuming that the Yorktown itself never did anything to garner the attention of people in the Federation. Our problem is that at that point we've seen only three designs in the motion picture era: Connie Refit, Miranda, Excelsior prototype, Excelsior production model and the Excelsior refit.
In 2293 when the Enterprise-B was launched that's only 7 years from NX-2000 to the Enterprise-B. And since the design of the E-B was so different than NCC-2000 you have to wonder, at that point and not knowing they would be the work horse of TNG, if there would really be that many Excelsior class ships around. And Excelsior herself was still in drydock in 2287, two years after it's first appearance. So while there may have been more on the planning boards or the beginning stages of being built there would be no Excelsior class Yorktown in 2286 to place old Yorktown's captain on.
As far as the public not caring about a hand me down being used for the name Enterprise I bet that there are those in Starfleet that would. Naval officers of our time are a very proud group in regards to the ships they serve. Let the current aircraft carrier USS Enterprise retire and then the navy slap the name on a tug boat and you betcha the public would care. I see the 23rd century being no different.