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"?:
And:
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:
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:
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:
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:
Later they enter solar system L-374:
But when they find the constellation in system L-374, and replay the Constellation's log:
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:
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.