In TOS it is mentioned several times that there are only 12 Starships in Starfleet. These are supposed to be the frontline vessels that protect outposts, colonies, planets and other smaller Starfleet vessels. In a couple of episodes it is established that no other Starships are in the area to assist the Enterprise. My thought is that there should be many more Starships to cover the immense amount of space just in our galaxy alone. What do you think?
Your statement that:
My thought is that there should be many more Starships to cover the immense amount of space just in our galaxy alone.
Implies that you think that the Federation has starships patrolling in other galaxies as well as a in our own galaxy. That is incorrect.
Our galaxy is extremely big, Very, very, very, very big. It has literally at least one hundred billion (100,000,000,000) star systems and possibly several times that many. It probably has hundreds of millions of planets with life and possibly many times that number.
But each of the major space powers in the Star Trek galaxy seems to rule only a few highly advanced and very populous planets - maybe tens, hundreds, or thousands of such highly advanced planets that a contribute a lot to the power of that government. So every space travelling society ever known in all of Star Trek combined would rule only a tiny fraction of our galaxy.
Thus the Federation's Starfleet needs only a tiny insignificant number of starships compared to the vast number of starships required to patrol the entire galaxy. In E.E. Smith's famous
Lensman series the governments of two galaxies fought increasing spectacular battles. At first they had fleets of millions of space battleship each, but the fleets got bigger.
But how large is the volume of space ruled by the Federation? How many starships are needed to patrol the volume ruled by the Federation, and does the federation have as many starships as it needs?
The Federation might include tens, or hundreds, or thousands of highly advanced and densely populated planets capable of building starships and other defense systems. And for each star system with such a planet, how many other star systems without advanced planets are in Federation space?
And another question is how many starships are needed to defend Federation space? Should the ratio be ten starships for every star system, or ten star systems for every starship? Or some totally different ratio?
Nobody ever said that there were only 12 starships in Starfleet during TOS. What was actually said was:
KIRK: Bridge.
CHRISTOPHER: Must have taken quite a lot to build a ship like this.
KIRK: There are only twelve like it in the fleet.
CHRISTOPHER: I see. Did the Navy
One question is did Kirk mean there were only 12 ships like the Enterprise including the Enterprise, or the Enterprise and 12 other ships like it?
Another question is did Kirk mean the entire Starfleet when he said "the fleet" or did he mean something like the Tenth Fleet when he said "the fleet"?
Another question is how similar to the Enterprise did a ship have to be for Kirk to say it was like the Enterprise? Would Kirk call any ship that was vaguely similar to the Enterprise "like it", or would Kirk only call ships that were very, very, very similar to the Enterprise "like it"? Were there many starships that Kirk didn't count as being like the Enterprise?
And was Kirk even telling the truth about the 12?
Another possible hint about the number of starships is in "Court Martial":
STONE: Stop recording. Now, look, Jim. Not one man in a million could do what you and I have done. Command a starship. A hundred decisions a day, hundreds of lives staked on you making every one of them right. You're played out, Jim. Exhausted.
So what percentage of the Federation's population did Stone consider to be "men", and how many such "men" were there?
Did Stone say that not one man out of a million had the basic personality needed to command a starship, or did he say that not one many in a million both had the right personality and also the right training from Starfleet Academy and experience in Starfleet to command a stasrhip?
In any case Stone claimed that the total number of Federation starships commanded by "men" was less than one millionth of the total number of "men" in the Federation. But of course there could have been billions or trillions of "men" in the Federation, thus making it possible for the Federation to possibly have up to thousands or millions of starships commanded by "men".
So there isn't much hard numerical evidence about the number of starships in Starfleet during TOS.