The 1701 Refit was destroyed in TSFS - before then it wasn't a museum piece (at least I don't recall the movies mentioning anyhing about that). It was just sitting in space dock and then SF decided not to repair it or put it back into action.
We don't know for certain if the 1701-A as seen in TVH was a completely new Constitution class ship or another rundown Connie that was refit... but I got the impression it was brand new that because of that it was also untested and had many problems.
The first book to tackle an explanation was "Mr Scott's Guide to the Enterprise" (non canon, of course). The author, Shane Johnson (now known as Lora Johnson) suggested, IIRC, that the
USS Ti-Ho was the first scratch-built Constitution II Class (now Enterprise Class) vessel. All the other ships in this class had been overhauled in the same way as the
USS Enterprise and other surviving Constitution Class TOS vessels.
Not long after its publication, Richard Arnold was talking about Gene Roddenberry's own theory at conventions and in "Star Trek Communicator". GR had not liked much of "Mr Scott's Guide to the Enterprise". Instead, he floated the idea that the ship unveiled as the
USS Enterprise-A at the end of ST IV had been a last-minute rename of the
USS Yorktown, taking the name from his own first Star Trek proposal. But then it was pointed out by fans that
we had just seen the captain (former tennis pro turned actor, Vijay Amritraj, as Captain Joel Randolph) of a
Yorktown reporting on his ship's disabling by the Probe, and hope to return to base with a solar sail, so there was no time for the new "-A" to be a refit of that particular ship. So again, more likely that the intended "Yorktown-A" was a scratch-built replacement for the fleet and the old Yorktown was due to be decommissioned. Hence his operative problems (mainly for comedy) in ST V.
Re the "20 years old" quote about the Enterprise of "ST III: The Search for Spock. This was a last-minute script correction. Starlog's ST III Official Magazine had a quote from an earlier transcript that actually said "40 years". Supposedly the "20 years" reference in the movie was to acknowledge the approach of the 20th anniversary of TOS, for which Paramount had big plans. (Any fan who saw "The Cage" and "The Menagerie" knew that the Enterprise was closer to 40 than 20. On a technicality, most of the ships parts were probably 20 years old, since it would have undergone a refit between Pike and Kirk's tenures.)