Just thinking out loud:
The only outright sexism I recall in TOS was in the episode I hate the most: "Turnabout Intruder"
But, the thing is, the one doing the complaining was Janice Lester, who, was obviously a crazy person who never got over Kirk dumping her (probably because she was a vengeful little nutjob long before she became a murderous vengeful little nutjob.)
But, it's been years since I saw that episode.
Maybe Janice Lester didn't make it in Starfleet because everyone else in it could tell she was a vengeful little nutjob and didn't want to let her anywhere near the command seat of a starship.
But the sad truth about vengeful little nutjobs is they don't know anyone else can tell that they've vengeful little nutjobs, so she had to find an excuse why she didn't make it in Starfleet and she blamed "Sexism" instead of herself.