There were actually some pretty sexist episodes of TOS. I wouldn't neccesarily say TOS as a whole was sexist, but there were some really shit moments that have horrified casual viewers when they decided to binge the series. Roddenberry was also a borderline creepy sex pervert and often worked some weird fantasies into his work. Reading some of his design notes for Phase II or TNG can illuminate this, especially in regards to the Deltan race. I wouldn't say he flat-out hated women though, that's just ridiculous.
TOS was a product of its time. He still had, in the pilot, Majel Barret placed as number one. Uhurra, which was extremely progressive for its time.
Female starfleet officers, judges, captains, admirals. You could infer that woman of the federation were treated equally as men....
Granted, Kirk liked fucking women and was slightly patronizing at times. Yes, the finale had a bizzare ending with Kirk getting possed by a woman and proceeding to have a neurotic breakdown. But that ending sucked and everyone hated it.
The main fault of TOS(in a modern context), the main dynamic did not include a woman. Most of the supporting cast are males. Hence, andocentric. But, as you said, far from women hating...
She also segued that criticism to TNG, which bent over backwards at times to show gender equality....
Whatever...
I think its bitching by Mulgrew because most fans view her as the worst captain. Since she has a massive ego(apparently), she can't possibly see why fans disliked her other than sexism.
Therefore, Star Trek needs even more female representation to fix its fanbase fulled of toxic masculinity....
I'm also going to go out on a limb and say she has never watched a single episode of 'Star Trek discovery'. But, she will comment on why fans must not like it...racism/sexism...