If you total up all the named characters from the show, you get at least a hundred. That's a fair portion of the crew. And yet, there are just five who are ratings out of 100 episodes. It's still statistically rare.
It sounds like you already have totaled up these named officers from TOS. Can you share this list as I'm curious as to who all the named officers are outside of the usual bridge crew and recurring officers.
That's a list of unnamed characters, not unnamed enlisted. That many of them are called "crewmen" doesn't mean anything -- the Lieutenants are called "crewmen" too.
As I pointed out in an earlier post - an officer can also be referred to as a crewman (in dialogue). The observation is that there are plenty of crewmen that do not have officer stripes that are unnamed. Since the story follows the bridge crew and not the lower decks crew then statistically you wouldn't see that many named enlisted personnel.
You'll note, too, there are examples of people without braid on their sleeves showing up at other times with a Lieutenant's stripe.
Did they jump from Ensign to Lieutenant or Sergeant to Lieutenant?
Starfleet has a robust cross training/job rotation and career development
You expect a lot from your crew! There's a difference between a Yeoman serving as a makeshift security officer (Tamura) and a Yeoman driving one of Starfleet's most sophisticated vessels...
See cross training comment above
EDIT: Another food for thought... there are examples of "officers" being referred to specifically. And in one instance "officers and crew" as separate groups. Wouldn't this tell you that if there are specifically "officers" that there are crew that are not officers?
"Turnabout Intruder"
SPOCK: The witness, sir. Bring on the witness. Let your
officers put the questions.
"The Savage Curtain"
KIRK: Mister President, may I present my
officers. Commander Spock, second in command, Engineering Officer Scott and Security Officer Dickerson.
"The Immunity Syndrome"
SPOCK: Personal log, Commander Spock, USS Enterprise. I have noted the passage of the Enterprise on its way to whatever awaits it. If this record should survive me, I wish it known that I bequeath my highest commendation and testimonial to the captain,
officers, and crew of the Enterprise. The finest starship in the fleet.
"Breads and Circuses"
CLAUDIUS: Order your
officers to come down.
"The Deadly Years"
STOCKER: Immediately, Mister Sulu.
All officers are to return to their posts.
"Catspaw"
KIRK: If we weren't missing two
officers and a third one dead I'd say someone was playing an elaborate trick or treat on us.
"The Return of the Archons"
Captain's Log. Stardate 3156.2. While orbiting planet Beta Three trying to find some trace of the starship Archon that disappeared here a hundred years ago, a search party consisting of two Enterprise
officers were sent to the planet below.
"The Menagerie"
SPOCK: Sir, I must point out that there are three
officers of command rank available. Yourself, Commodore Mendez, and Captain Christopher Pike.
"Court Martial"
Captain's Log, Stardate 2948.9. The
officers who will comprise my court-martial board are proceeding to Starbase Eleven.