The problem is that nothing in TOS or TAS counteracted Lester's claim.
Perhaps not explicitly, but nothing reinforced it either. And Number One commanding the ship in Pike's absence certainly comes close.
In any case, it's not a problem, because this is fiction. We're not watching a documentary beamed from the future. We're not required to treat every last line of dialogue or frame of film as some immutable truth. We're allowed to reinterpret it, to ignore its mistakes. Roddenberry himself wanted fans to do that. He admitted in later years that TOS contained mistakes and bad ideas that shouldn't be considered binding. When he had the Klingons redesigned in TMP, he asked fans to accept that they'd always looked that way and TOS just hadn't depicted them correctly. His introduction to the TMP novelization adopts the pretense that TOS was a 23rd-century dramatization of Kirk's "real" adventures and was "inaccurately larger-than-life" compared to the real thing.
So there is no problem. If we can pretend that the wrinkly space art in the side screens of the bridge was actually an approximation of video screen images, or that the crude marionettes of Sylvia and Korob's true forms with visible strings were just an inaccurate representation of something more alien, then we can pretend that TOS's portrayal of gender roles and relations was another inaccuracy resulting from the '60s attitudes through which the depiction was filtered. These are just stories, and stories are filtered through their tellers. But we're allowed to refilter them through our own interpretations.