You know. I've finally realized something.
After reading through the thread again, I finally realized that I made a mistake in my original post. Instead of trying to offer a reasonable explanation for the situations the OP described, I should have responded by asking the OP this question:
What exactly is your problem with seeing more than one ship of the class in the same area at the same time? Seriously. Starfleet builds, owns and operates the ships. If they have no problem with a bunch of them in one spot, why do you?
By the way, I invite everyone in the thread to answer this question, because most responders have engaged in high impact mental gymnastics like trying to redefine the concept of "class" or trying to prove a ship is a different class depending on whether some widget is mounted on the right or on the left on ships that look exactly the fucking same. Why? What grievous sin has Starfleet committed by deploying more than one Constitution class ship to a single area of operation?
Why did they send a Constitution class to aid other Constitution classes? Are you kidding? Who better to send to render assistance than a crew that's intimately familiar with the design of the ship in distress? Why send several of their "elite" ships to one battle or training area? Well, if one "Elite" ship is awesome, then two "elite" ships are twice as awesome, and four "elite" ships is twice as awesome as two, and about six or seven "elite" ships would be fucking bad-ass. Why not send as many as you can spare? Maybe one of them would get damaged? Isn't the whole point of "safety in numbers" the idea that it lessens the chance that any one unit will be damaged beyond repair?
The only thing I can figure is that OP and others are sufffering from TV Producers Disease, a psychological ailment that makes them feel like they've failed if they don't provide something new to see every episode. It's a disease common to fanfic writers who can't think of beginning a story withour creating a whole new class of super starship out of whole cloth. In this instance, the disease makes viewers lament the "limitations" of effects technology when TOS was produced. In fact, those limitations make TOS more realistic, because they show ships the way they're procured in real life - in batches to save cost and resources - as opposed to TNG, whose producers would kitbash and improvise designs to death just produce ship designs you'd maybe see once, just 'cause they were new.
Somebody help me understand. We're supposed to be treating this discussion in universe, so tell me in universe how the deployment of Connies is somehow detrimental to the Federation.
Why?
Because space is really big. As Douglas Addams says:
Space is big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road to the drug store, but that's just peanuts to space.
https://www.brainyquote.com/quotes/douglas_adams_164151
And ships like the
Enterprise are rare. Nobody knows how many Starfleet starships there are. Nobody knows how many Constitution class starships there are. Nobody knows how many ships there are that look a lot like the pliot or pruduction versions of the
Enterprise from the outside and from a distance.
But what we do know is what Kirk said in "Tomorrow is Yesterday":
CHRISTOPHER: Must have taken quite a lot to build a ship like this.
KIRK: There are only twelve like it in the fleet.
And if the Federation has maybe a hundred main member star systems, and claims hundreds or thousands of other star systems, and is exploring a volumeof space that includes many thousands or millions of star systems (which owuld still be a tiny and insignificent proprtion of the MIlky Way Galaxy), having only 12 or 13 top of the line ships is a bit of a problem.
So maybe each of the possibly dozens of main member star systems in the Federation has a mighty system defense system, and the home world of every member species in each such system has a mighty planetary defense system even stronger than the defense system for the entire star system.
I suppose that in science ficioin an ideal planetary defense system would be like that of the planet Onlo or Trallis IX in E.E. Smith's
Second Stage Lensman. One galactic government sends its fleet of millions of space battleships to invade another galaxy, and the second galaxy sends a fleet of millions of space battleships to stop them. After a tremendous space battle, the defending fleet is destroyed and the invading fleet continues their invasion. But despite their fleet being powerful enough to defeat millions of space battleships, the invading fleet doesn't even try to attack Onlo, and its commanders can't think of any method to attack it.
Fortunately space fleets in
Star Trek only have tens, hundreds, or at most thousands of space warships, and they are much less powerful than the space batleships in the
Lensman series. So a planetary defense system for a major planet in
Star Trek can get by with only a tiny fraction of one percent of the total power of the best planetary defense systems in the
Lensman series.
But there are many colony planets, mining planets, scientific outposts, etc. in the Federation without the massive defense systems of the main planets. Presumably the vast majority of systems in Federation space need mobile protection by Federation space warships from enemy space warships.
So maybe the exploring scout ships and survey vessels and science vessels and starships are only a minority of the ships in Starfleet. Pehrps they are outnumbered many times by the dedicated warships which are kept in various war fleets in strategic locations in Federation space, ready to respond to attacks and invasions.
The size of the Federation, and the size of the area where starships explore, is not known. But it seems very probable that the "12 like it" that Kirk mentioned should be scattered amoung tens, hundreds, or thousands of star systems each, with the nearest other one of the 12 being many light years away, probably days, weeks, or months of travel away.
But if Starfleet has many times as many major exploring ships on the frontier as Kirk's 12, those major ships would be more common and it would be more common for such ships to meet each other. But it would be rare for both of the meeting ships to belong to the minority "12 like it". So that is a good reason to assume that most ships met by the
Enterprise, even large starships, are not of the same class and are not included in Kirk's "12 like it".
So possibly the Federation is shaped like a sphereoid, and there are 10 or 20 equal sections on its frontier surface, and there are 10 or 20 volumes of space extending out from each section of the Federation border. And maybe each of the 10 or 20 volumes has a set of about a dozen powerful starships exploring that volume. So maybe one section has the Constitution class starships, and another section as the Xantynol class starships, and another section has the Hr'afnoth class starships, and so on.
Or possibly there are about 100 constitution class starships, but "there are only 12 like it in the fleet", meaning the fleet of constitution class starships assigned to exploring that particular volume of space, and not meaning the entire Federation Starfleet.
Or possibly there are only 12 starships in Starfleet, and they all explore the same narrow region of space at the same time. And when they have explored a set distance beyond the frontier in that direction, they are all reassigned to explore another comparatively small region of space at the same time
And that would explain why the
Enterrprise travels to stars that are in several widely different directions from Earth, and visits different starbases, and why KIrk is subordinate to different superior officers, during TOS. The
Enterrprise is sometimes reassigneed to different regions of space. And possibly all the other ships in "the 12 like it" are also reassigned to those same volumes of space at the same time as the
Enterprise. That would make encountering them somewhat more statistically probable.
And that is the kind of speculation which is necessary to make sense out of the
Enterrprise meeting several other ships which look similar to it on the outside, and which might possibly - though not certainly - belong to the same class of ships, and which might possibly - though not certainly - be among the few, the very few, "12 like it" that Kirk mentioned.
So I hope that now you understand why some
Star Trek fans are unwilling to accept the idea that all the ships that look a lot like the
Enterprise actually belong to the same class of ships or actually are amoung the "12 like it".