I don’t get where you’re hearing “male privilege” as a feminist response to men’s problems. Male privilege doesn’t mean that men don’t have any problems, and I don’t know of any feminist who believes it does. (It does mean that men shouldn’t assume that the problems of sexism are the same for women as for men, or that it’s okay for men to try to divert discussions of women’s problems with sexism to talk about men’s problems instead, which is the sort of context in which I generally hear “male privilege” applied to men talking about their problems.)
In terms of traditional sexism, there isn’t really any culturally embedded “female privilege”, just as in terms of traditional racism, there isn’t really any culturally embedded “black privilege”. The residual privilege left over from traditionally sexist and racist societies still so disproportionately benefits men and whites, respectively, that it’s hard to know what “female privilege” would even mean.
If it means “there’s a culturally embedded sexist expectation that certain kinds of irresponsible behavior should be tolerated in women where they wouldn’t be tolerated in men because women are innately inferior so consequently less is expected of them”, that certainly exists, but it’s not actually a counterpart to “male privilege”. Any more than the fact that small children can get away with crying and screaming in public because they’re just kids counts as “child privilege” on a par with “adult privilege”.
Toxic femininity, as we discussed at length in the previous thread on this topic, is in fact a subject of public discourse. I definitely wish people would talk more about the ways that traditional femininity is used to disempower and infantilize women under the pretense of empowering and fulfilling them—in other words, toxic femininity—but it’s not as though it never gets talked about at all.