As this makes clear (and one of the commenters points out) the main reason people (including many feminists, I believe) don’t like the Supergirl trailer is that the hero is female.
BTW, there is just as much “bubbly nonsense” in male superhero shows but that gets a pass because it’s ok then.
Men won’t like the show because it’s got a female lead and because she’s not there to be eye candy.
Feminists will torch the show because it doesn’t hit every feminist talking point, and because it’s not dark and brooding, and because the character is realistically awkward and not some insta-badass.
A character that is sincere, thoughtful, femme, bubbly and not quite comfortable in her own skin? Yes, please. (I suspect a lot of feminists also dislike her because she is in fact very feminine. Internalized misogyny is pervasive.)
The show will last one season, I predict.