It is very, very normal for ships to have multiple identities in their lives, and if they get selected as museum pieces, to be restored to the most significant or emphatic point of their careers.
So "Commissioned as Enterprise-A, renamed when Enterprise-B commissioned; eventually paid off, handed over to the museum, and repainted as Enterprise-A" is very, very reasonable and almost as likely as "commissioned as some other ship, renamed as Enterprise-A for Kirk, and paid off when Enterprise-B commissioned." (I personally think the latter is more likely because it explains all the problems in TFF.)
So here are some scenarios:
1. The
Enterprise-A started life, say a decade before as the
U.S.S. Michigan NCC-1785, was later scheduled for decommission but instead was hastily renamed and re-registered as the
U.S.S. Enterprise NCC-1701-A as a tribute to Kirk, with the understanding that he would only be commanding it temporarily until the
Enterprise-now-B was finished being constructed, seven years later. The A suffered from systems failures (possibly the reason why it was going to be decommissioned), but those failures were fixed by the time the ship was given its final mission to escort Gorkon to a peace conference, which was also when Starfleet gave the notice that the ship was to return to Earth to be decommissioned to make way for the B. Then the ship was sent to the Starfleet Museum and kept its new registry.
2. The
Enterprise-A started life, say a decade before as the
U.S.S. Michigan NCC-1785, and was later hastily renamed and re-registered as the
U.S.S. Enterprise NCC-1701-A as a tribute to Kirk, who would be commanding it indefinitely. The A suffered from systems failures for some reason, but those failures were fixed by the time the ship was given its final mission to escort Gorkon to a peace conference, which was also when Starfleet gave the notice that the ship was to return to Earth to be decommissioned to make way for the B. Then the ship was recommissioned back to the
Michigan with its original registry, continued to serve in Starfleet for several more years/decades, and then finally decommissioned and given its temporary
Enterprise name and registry for placement in the Museum.
3. The
Enterprise-A was a brand-new ship just commissioned right after the NCC-1701 was destroyed over Genesis. For some reason, this brand-new ship suffered from systems failures right from the get-go, but those failures were fixed by the time the ship was given its final mission to escort Gorkon to a peace conference, which was also when Starfleet gave the notice that the ship was to return to Earth to be decommissioned. Because the ship was only seven years old, the only logical reason why the ship would have been decommissioned so early was to make way for the B. Then this seven year old ship was sent to the Starfleet Museum.
4. The
Enterprise-A was a brand-new ship just commissioned right after the NCC-1701 was destroyed over Genesis. For some reason, this brand-new ship suffered from systems failures right from the get-go, but those failures were fixed by the time the ship was given its final mission to escort Gorkon to a peace conference, which was also when Starfleet gave the notice that the ship was to return to Earth to be decommissioned. Because the ship was only seven years old, the only logical reason why the ship would have been decommissioned so early was to make way for the B. But instead, this seven year old ship was given a new name and registry, say the
U.S.S. Pathfinder NCC-2025, continued to serve in Starfleet for several more years/decades, and then finally decommissioned and given its original
Enterprise name and registry for placement in the Museum.
So while I get that in real life, ships can have multiple identities, Star Trek isn't real life. I tend to go by Occam's Razor in these instances. Taking into account the on-screen evidence, such as the ship's system failures, Scotty calling the ship 'new' in dialogue, the relatively newer TNG-style interiors and the newer bridge module, the fact that the crew seems completely surprised by the announcement of the decommissioning, and the fact that the B was commissioned not soon after, the simplest answer seems to be that the
Enterprise-A was a brand-new ship that was decommissioned after only seven years of service, and was sent directly to the Museum where she still exists in 2401. It's not the most logical answer, but it is the simplest. Kirk does mention that 'this ship and her history will shortly become the care of another crew,' but that was clearly just a reference to TNG, and not literally another crew that will take over the
Enterprise-A.