It would be interesting to base a "worldview" of the STAR TREK Universe strictly on the content of TOS exclusively, when we still had not seen any other "class" of "starship" other than the one of which the Enterprise was apparently a member. This would imply that there is only one kind of starship and that's it for the fleet. (It would be interesting to reconcile the names of all the starships mentioned in TOS against Kirk's brag to Captain Christopher, USAF, in "Tomorrow is Yesterday", about how "there are only twelve like it in the fleet".)
Let's see what kind of list we would come up with...
U.S.S. Valiant ("A Taste of Armageddon")
Antares? ("Charlie X")
Astral Queen? ("The Conscience of the King")
The list of ship names (apparently all Federation starships) from Commodore Stone's wall chart seen "Court Martial":
(The Memory Alpha article on Greg Jein's "The Case of Jonathan Doe Starship"
also contains other references.) It is worth mentioning that Kirk also mentioned the U.S.S. Republic, registry 1371, in his deposition.
Archon? ("The Return of the Archons"; Kirk does refer to the Starship Archon as missing)
"Constitution class" is shown on Khan's computer display in "Space Speed"; the "technical manual" schematic says:
PRIMARY PHASER L, R
STAR SHIP MK IX/01
While we could give ourselves a Mulligan either way on this, either dismissing it as not clearly seen or accepting it as the first canon mention of the Constitution class starship, it does bear mentioning at the very least.
Unit XY-75847? (mentioned in "Errand of Mercy")
U.S.S. Carolina ("Friday's Child")
U.S.S. Constellation ("The Doomsday Machine")
U.S.S. Farragut, U.S.S. Yorktown ("Obsession")
U.S.S. Intrepid (remastered image in "Court Martial" and mentioned in "The Immunity Syndrome")
Horizon? ("A Piece of the Action")
"The Ultimate Computer": Wesley's task force...
U.S.S. Exeter ("The Omega Glory")
U.S.S. Defiant ("The Tholian Web")
If we ignore Commodore Stone's wall chart, I still count at least 19 ships. (Of course, that's a total count of the entire TOS roster, including those encountered a year or two after Kirk's "Tomorrow is Yesterday" brag to Capt. Christopher.)
Discounting XY-75847, Archon, Astral Queen, Antares, and Horizon, as not clearly being identified as Federation starships (I strongly feel that the way Archon and Horizon were mentioned confirms they were
, but that's just me) would bring the total starship roster to 14.
Then, of course, there's the matter of Stone's wall chart and the mention of a Constitution class phaser used in "Space Seed" and "The Trouble with Tribbles" to quibble over. If there is a Constitution-class ship, then there is likely to be a U.S.S. Constitution. Greg Jein's work seems to support this. That would bring the total number of Federation starships to 15.
If there were only a maximum of 14 to 19 "Star Ship Class" vessels in the entire Federation at the time of "The Tholian Web", this would certainly explain why being a captain of one would be regarded as extraordinary.
It also would not square with TAS, TWOK, TVH, etc.