So FTL travel is my "personal worldview" of Trek. Then explain warp is supposed to work that I'm missing....
FTL is not one speed, it is a fast range of possible speeds ranging from the speed of light to infinite. And thus it is an infinite range of speeds.
But certanly there is no evidence in TOS that starships have a warp factor with infinite speed. And if a ship can not reach infinite speed there will always be some plots involving that ship which are impossible because of matters of insuficient speed.
What was said in "By Any Other Name"?:
KIRK: What do you want?
ROJAN: Your ship, Captain Kirk. It will serve us well in the long voyage that is to come.
KIRK: Voyage? Where?
ROJAN: To your neighbouring galaxy, which you call Andromeda.
KIRK: Andromeda? Why?
And:
KIRK: What's the point of capturing my ship? Even at maximum warp, the Enterprise couldn't get to Andromeda galaxy for thousands of years.
ROJAN: Captain, we will modify its engines, in order to produce velocities far beyond the reach of your science. The journey between galaxies will take less than three hundred of your years.
Spock Fascinating. Intergalactic travel requiring only three hundred years. That is a leap far beyond anything man has yet accomplished.
The Andromeda Galaxy is about 2,500,000 light years from Earth. Assuming that Kirk meant it would take the Enterprise between 1,000 and 10,000 yeas to travel that far at maximum speed, the maximum speed of the Enterprise would be somewhere between 250 and 2,500 times the speed o flight.
In "the Omega Glory" the Enterprise finds another starship in orbit around a planet:
SULU: It's the USS Exeter, sir.
KIRK: Try to contact her, Lieutenant.
UHURA: Aye, sir.
KIRK: The Exeter. she was patrolling in this area six months ago. I hadn't heard of any trouble.
So Starfleet lost contact with the Exeter 6 months or less earlier, and Kirk apparently hasn't been ordered to find her, but just accidentially found her. Suppose instead that Starfleet command ordered Kirk to go looking for the Exter 6 months ago and the Enterprise has been travelling straight to that solar system for 6 months at top warp speed.
In that case the Enterprise could have travelled about 125 to 1,250 light years from its old position straight to the posiiton of the Exeter. But instead the Enterprise didn't know that anything had happened to the Exeter and was travelling from star to star in various directins at varius speeds on various other missions during those six months.
So the initial distance between the Exeter and the Enterprise when tthe plague struck the Exeter should have been much less than 125 to 1,250 light years.
In "The Tholian Web" they are looking for a missing starship:
Captain's log, stardate 5693.2. The Enterprise is approaching the last reported position of the star ship Defiant, which vanished without trace three weeks ago. We are in unsurveyed territory.
Exactly three weeks is exactly 7 days. Assuming that Kirk would round up or down to "three weeks" if the actual time span was somwhere between 17 and 025 days, the Enterprise should have traveled no more than 4,250 to 62,500 light days, which is 11.635 to 171.115 light years, toward the Defiant in that time, and possibly a lot less.
In "The Immunity Syndrome" Starbase 6 tries to contact the Enterprise, and Spock suddenly feels the death of the Vulcans aboard the Intrepid. And then:
STARBASE [OC]: You will divert immediately to sector three nine J.
KIRK: Sir, the Enterprise just completed an exhausting mission. We're on our way in for R and R. There must be another starship in that sector.
STARBASE [OC]: Negative. This is a rescue priority. We've lost all contact with solar system Gamma Seven-A, which the Intrepid was investigating. And we've just lost contact with the Intrepid. Report progress.
KIRK: Order acknowledged. Kirk out. Mister Kyle, you heard the order. Set course for Gamma Seven-A, warp five.
KYLE: Aye, sir.
CHEKOV: Captain I have just completed a full long-range scan of Gamma Seven-A system. It is dead.
KIRK: Dead? It's a fourth magnitude sun. There are billions of inhabitants there.
There is no statement about how long it takes for the Enteprise to reach the Gamma Seven A system. So I would guess that it takes the Enterprise between one hour and one week to travel the distance, which should be no more than 0.028 to 47.912 light years.
In "The Doomsday Machine" the Enterprises earches for the starship sending out a distress signal:
ULU: Sir, we're now within the limits of System L-370, but I can't seem to locate
SPOCK: Captain, sensors show this entire solar system has been destroyed. Nothing left but rubble and asteroids.
KIRK: But that's incredible. The star in this system is still intact. Only a nova could destroy like that.
SPOCK: Nonetheless, Captain, sensors show nothing but debris where we charted seven planets last year.
KIRK: Continue a search pattern.
Later they enter solar system L-374:
SULU: Entering limits of System L-374, sir. Scanners show the same evidence of destruction.
KIRK: Every solar system in this sector blasted to rubble and still no sign of the Constellation. Matt Decker's in command. What could have happened to him?
SPOCK: Captain, the two inner most planets of this system appear to be intact.
But when they find the constellation in system L-374, and replay the Constellation's log:
DECKER [OC]: Captain's log, stardate 4202.1. Exceptionally heavy subspace interference still prevents our contacting Starfleet to inform them of the destroyed solar systems we have encountered. We are now entering system L-374. Science Officer Masada reports the fourth planet seems to be breaking up. We are going to investigate.
So the planet killer finished destroying tthe fourth planet and then destroyed the third planet in the interval.
Matt Decker is still alive, so life support has not yet totally failed, and he hasn't committed suicide yet, but he is still very traumatized over the fate of his crew. That gives me the impression that the third planet was destroyed no more than a few days ago. So I would guess that the Enterprise was no more than a week's travel, or 47.912 light years, from system L-374 when the Constellatin sent out the distess call.
And a star system is mentioned later:
SULU: It's veering off, back on course for the next solar system. The Rigel colony, sir.
SPOCK: Evidently programmed to ignore anything as small as a ship beyond a certain radius. We'll maintain a discreet distance and circle back to pick up the Captain.
DECKER: You can't let that reach Rigel. Why, millions of innocent people would die.
SPOCK: I am aware of the Rigel system's population, Commodore, but we are only one ship. Our deflector shields are strained, our subspace transmitter is useless. Logically, our primary duty is to survive in order to warn Starfleet Command.
So star system L-374 seems to be beyond Rigel as seen from Earth. Since Rigel is about 860 light years from Earth, that indicates that exploring starships should lbe operation on the surface of a sphere of space with a radius of about 860 light years - or beyond it.
A sphere with a radius of 860 light years would have a surface area of 9,294,087.71 square light yers. If 12 starships are equally spaced on the surface of the sphere each would have an area of 774,507.3 square light years to itself. That are would equal a square flat surface that was 880.06 by 880.06 light years square, or a round flat surface that had a radius of 496.042 light years, wrapped on the surface of the sphere.
And that is not allowing for how much more or less beyond the 860 lightyear radius various starships might be at any one time.
There is absolutely no identification of starship classes in TOS. Nobody ever mentions how many starships Starfleet Command has.
Kirk does say that there are only 12 starships like the Enterprise in the fleet. If that is interpreted as meaning there are only 12 starships of the Enterprise's class in Starfleet, then Starfleet needs to have other classes of starships for.the Enterrpsie to meet as many starships as it does.