I think it's fair to say that nearly all shows written in the 60s had the underlying assumption that women were not fit for leadership or technical roles and TOS is no exception.
This statement ignores quite a number of female science specialists and the obvious command role of the Number One character.
Only
Turnabout makes a point about there not being female starship captains. And, strictly speaking, they were likely talking about one elite class of 12 vessels out of who knows how many thousands of Star Fleet ships. I seriously doubt there were no female captains among so large a fleet.
But you also can't ignore these:
-- Kirk's line in "Elaan of Troyius": Mr. Spock, the women on your planet are logical. That's the only planet in this galaxy that can make that claim."
-- The idea of the "Russian brides" business being alive and well in the 23rd century in "Mudd's Women". Every woman needs a husband, and every miner needs a woman. After all, Eve knew how to clean pots and pans. Very useful. Indeed, Kirk never seems too upset at the idea that Mudd is trafficking wives like commodities. It's just a business.
-- Nomad scans Uhura and concludes she is defective, with chaotic thinking. Spock responds to this by telling Nomad that "unit" is a woman. Nomad says she's a mass of conflicting impulses.
-- "She's a woman, all woman. One day she'll find the right man and off she'll go, out of the service," McCoy said this about Lt. Palamas in "Who Mourns for Adonais?". Why?
-- In "Shore Leave", fantasies seem to be fulfilled. Tonia's? Being a fairy tale princess. "A lady to be protected and fought for," as she said.
-- In "Wolf in the Fold", Spock believes the creature only attacks women because they are more easily terrified and generate more horror than men.
-- Even with Number One as second in command, Pike says in "The Cage" that he just can't get used to a woman on the bridge. When he noticed the yeoman he's referring to overheard the remark, he tells her, "No offense, lieutenant. You're different, of course." WTF does that mean?
-- In a similar vein, what kind of professional (and progressive) atmosphere is it when Kirk remarks to McCoy about being upset at having a woman yeoman, and McCoy responds, "What's the matter, Jim? Don't trust yourself?"
Yes, TOS had women lawyers, women officers (of course, not starship captains, as that wasn't allowed in the 23rd century), women diplomats, women scientists, and women aliens in positions of authority on their planets. But the other things were there, too. And today, they are particularly cringe-worthy to watch (and probably were then too, to a lot of progressive people).
I will say that in "Turnabout Intruder", Kirk does at least admit to Janice that it's not fair she can't be a starship captain because she's a woman. Then again, in the last line of the last ever TOS episode, Kirk says her life could've been as rich as any
woman's. What an odd distinction to make. Not as rich as any
person's? Or at least,
anyone's? Is the implication that what defines a rich life for a human in the 23rd century is still differentiated by sex?