Different era. They totally changed the quadrant designations by the time those ships came around. I don’t think they used that “only ship in the quadrant” nonsense since the TOS films anyway (with TFF being the last to do so maybe?). It’s become an argument of apples vs hand grenades now. The concept of the one isn’t compatible with the concept of the other.
Doing a search of the transcripts on Chakoteya.net, I find that, contrary to popular belief, the only use of the verbatim phrase "the only ship in the quadrant" was in
The Wrath of Khan.
The Final Frontier had Kirk say "There must be other ships in the quadrant," to which the admiral replied "Other ships, yes, but no experienced commanders."
The only reference I find in TOS itself is "Journey to Babel," where Spock says, "Starfleet records no authorised vessel in this quadrant except ours." But that's in the context of trying to identify the enemy ship, not a justification for the
Enterprise's assignment. The only other similar reference I can find is in TAS: "One of Our Planets is Missing": "Starfleet Command has sent the Enterprise to investigate as we're the only vessel in the vicinity of the phenomenon." For what it's worth, "Wolf in the Fold" said that Argelius II was the only spaceport in the quadrant.
So it seems this is a "Beam me up, Scotty" situation, something that's assumed in the lore to be a recurring Trek cliche but really isn't found in TOS itself.
Leaving aside the "only ship" bit, though, it's true that TOS used "quadrant" in a wealth of inconsistent ways that didn't seem to fit the idea of 1/4 of anything. It was as if the word had outgrown that literal meaning and people had just come to use it generally to mean a portion or segment of a thing. There were even references like "quadrant 904" and "quadrant 448" and "quadrant 779X by 534M."