I know. I grew up watching Star Trek, and own all 12 movies. Technically, you are correct. It's just easier to say "Enterprise-A" than it is to say "the refit Enterprise from ST I, ST II, and ST III". And yes, the -A was just a re-use of the studio model from the first 3 movies, and I see your point.
Now let's dive into canon itself: The Enterprise A, while it is just a re-use of the studio model, is canon in the Trek universe. Gene Roddenberry actually addressed the relative short life of the Enterprise A. According to him, the -A was actually not a newly constructed starship, and was at one time the USS Yorktown (as a nod to the to the ship's name in the original TV pitch for the original series). The ship was supposedly taken out of service, but because of the destruction of the original Enterprise, and that it is ridiculous to belive that Starfleet could build a ship in the short amount of time (ie Star Trek IV), the Yorktown was renamed and pressed back into service as the Enterprise-A. So, Rob, in a way, you are right. The short life-span of the Enterprise -A is due to the fact that the ship had been in service for several years prior to it's re-name, and if you think about it, it's assigned missions in the (terrible) ST V, and the (fantastic) ST VI, kind of reflect that it is not on front-line anymore. In ST V, it's mission is to rescue diplomats from a renegade Vulcan. Not exactly a job for a ship-of-the-line, provided all goes as planned. And in ST VI, it's job is to escort the Klingon representatives to a peace conference. Had all gone as planned, the mission is more ceremonial in nature. Being the most famous names in Starfleet, Kirk and the Enterprise were naturally selected, instead of the state-of-the-art Excelsior. The 1701-A is canon, and therefore it's own separate starship. But your "rename" argument is totally valid.
"Mr. Scott's Guide to the Enterprise", published shortly after ST IV was released, suggests more-or-less, the same idea: it was a ship (USS Ti-Ho) used as a testbed for the traswarp project, alongside USS Excelsior, and was just renamed to Enterprise. But it can be easily argued that this is not canon, so take that as you will.
Sorry, but I love getting the opportunity to discuss Star Trek, and I totally nerd out a little...OK, maybe more than a little. lol. Not many people I know are that into Star Trek as I am.