Since I've had Kirk's final log entry memorized since 1991, let's see what it suggests about the E-A:
"Captain's log, stardate 9529.1 This is the final cruise of the Starship Enterprise under my command. This ship, and her history, will shortly become the care of another crew. To them and their posterity do we commit our future. They will continue the voyages we have begun, and journey to all the 'undiscovered countries,' boldly going where no man-where no one-has gone before."
I think the issue of the ship's decommissioning stems from one poorly-written segment of an otherwise beautiful bit of writing. "This ship" was likely intended to refer to the legacy of the Enterprise name, not the E-A herself. Even if there was no buzz or indication that there would be an Enterprise-B, Kirk would have known (and anyone else for that matter) that the addition of a suffix meant that there would likely be more ships named Enterprise.
"This ship, and her history" could have been better rendered with something along the lines of "The legacy of the Starship Enterprise will shortly become the care of another crew." This would avoid singling out the E-A in particular and suggest what was intended with TUC, namely, passing the baton; this is most obvious is the line "where no man-where no one-has gone before," i.e., the TNG wording of the Starfleet Charter.
I take this minor problem of association to be at the heart of this argument. I also think the Enterprise-A was in fact a brand new ship though with a different name. She hadn't launched yet and was given the new commission of Enterprise 1701-A.
I also don't think she was either recommissioned for further use after repairs OR put in a Starfleet Museum to make way for the "next generation."
And let's get one thing straight: Captain James T. Kirk was a hero, along with his crew. If anyone at Starfleet hated/disliked him as much as some seem to indicate in this thread, then they were just jealous losers who wanted the glory for themselves.
I prefer to think they were seen as legends. The E-A was a gift, albeit one that had all the problems of a brand-new vessel with experimental tech (going on the Guide to the Enterprise) and was in no way a "punishment."
And it's clear to everyone assembled at the end of Star Trek IV that demoting Kirk was an honor and a reward, not a punishment.
Finally, I think Kirk and Co. went on various missions aboard the E-A, but were also doing other things as well. Kirk did bring a bag to his quarters, indicating he didn't live aboard the ship. However, I think Spock did spend a lot of time there, maybe even live there, going by the painting in his quarters and other decor.