What is harder to square away is the ship being called Enterprise. Sure, you might not include all the Enterprises ever in your rec room or briefing room, and Scotty specifying the 1701 registry helps explain why the computer didn't load Trois ENT fan film programme, but they have been pretty specific the E-D was the fifth starship to bear the name, and that pattern continued on the dedication plaques. The Department of Temporal Investigations also seemed not to include the NX01 in a context which could certainly have included it (time travel).
In short, naming the ship almost anything else would have made the story fit very plausibly into Trek's past.