I realize they trie to create different characters and personalities for the show, but I don't know that I have not noticed the man-hating sexism of Charlie, but I have observed that they have tried to make her this archetype lesbian nerdist and have gone a little bit overboard with the ' I am a lesbian and I will hit on every woman that I see'. I think most lesbians would disagree with this characterization.
I think the problem with Charlie is that she has no ties to hunting like Jo and Ellen and the character has not really contributed anything to hunting. She would be good as a tech specialist helping the bros do detective work, but the damsel in distress role has gotten old.
Charlie herself is not that sexist but her episodes often portray sexism.
1) All her episodes display women=superior, men=inferior.
It's ok if it's just one or two episodes but it's all of her episodes.
2) All of her episodes never has a male victim survive. I wouldn't count Sam or Dean since they're the stars of the show. All male victims always end up dead. To make it worst (in terms of sexism against men), there are a couple of female survivors.
3) She thinks men of letters is sexist but loves the sound of women of letters.
Yeah, real feminist double standard right there.
Now, feminists would say Supernatural is sexist because it always has female damsels forgetting that in every season, a vast amount of casualties are also male.
Aside from her sexism, she's also a poorly-written character.
1) As you said, she never had background in hunting. I missed the times of Jo wherein if your experience is not enough, and you go hunting, you're better off committing suicide.
Charlie did it and alone. This destroys my idea of hunters. It makes her very unrealistic.
2) Her last episode: Wizard of Oz was stupid imo. It wasn't comedy, it was a farce. It was the worst episode in the history of Supernatural. I almost quit watching Supernatural after that. It's a good thing the episode after that was very good.
3) As a character, she seems to have no weak points. All her episodes are just about people telling her how good she is. This makes her boring and just a really bad character overall.
Now, if they kill her or never show her ever again, I would be very happy.
Charlie is both an offensive sexist feminist and a poorly-written character.