Actually, she says "Your world of starship captains doesn't admit women."Lester clearly states the Starfleet doesn't "admit" women, which would suggest she means in a professional sense, not personal.
Exactly. Look at her very specific choice of words: YOUR world. Kirk's world. His own, personal world. He can't have a lasting relationship with ANY woman. That's what she meant.
I mean, it's not as if the perfect utopia Gene Roddenberry was trying to create would be so sexist as to not allow female captains, would it?
Again, how did TOS do during the same period?