I think it's more that the specific phrase in TWOK was adapted as shorthand for all the times the Enterprise was the only ship available to handle a giant space amoeba/runaway doomsday machine/ancient Earth probe that's transcended the bounds of reality.
Yes, but I looked for phrases like "only ship" and "only vessel" and didn't find the abundance of references that fan myth presumes to exist.
Really, though, the whole idea in TOS was that the Enterprise was automatically the only ship in the immediate vicinity. That was the whole point -- that it was patrolling the far frontier, protecting remote colonies and outposts because there was nothing else that far out that could protect them. It was built into the premise, and thus didn't have to be called attention to, nor was it an implausible excuse in those circumstances. It didn't become a problem until the movies, which started hanging around Earth and the Federation core worlds more, so that claiming the Enterprise was the only available ship to address, say, a giant alien energy cloud heading toward the capital of the entire Federation made no sense at all.