The whole Yorktown story was nothing more than Roddenberry's vanity speaking because that's what the ship was named in the original TOS draft.
I'm inclined to disbelieve it for the simple reason that, as somebody already said, the actual Yorktown was in the same movie and likely had a full crew aboard (the captain was putting all "non-essential personnel" into stasis when the power went out).
Another possibility I read about somewhere suggests the E-A was put together from salvaged components from various other ships. Since the Constitutions and Mirandas (and maybe even more classes) both were built from the same basic components, there could reasonably have been enough spare parts hanging around to make up a whole new ship with not a ton of effort.
This would also explain why so much of the ship's interior design doesn't seem to match. The corridors have an E-D aesthetic, while the bridge looks more like the original E, and the shuttle bay is tiny, much smaller than it should be.