This is my stand on sexism. Society shouldn’t be built upon the leadership of a person based on their sex, male or female. A society whose authorities are justified purely due to their sex, is a society that doesn’t optimally extract the potential out of its members.
Furthermore, the notion of a matriarchy is as sexist as a patriarchy is, because even when the upper hand is given to females purely because they’re females, it’s the same sexism that exists when the upper hand is given to males because they’re males and nothing more. Sexism is applied to both sexes, and by choosing matriarchy over patriarchy solely due to the leading figure’s sex, sexism is applied nonetheless.
What would therefore improve social construct is to not put people in leadership positions due to their sex, but, rather, do so in accordance to their individual merits, because even though there are differences between the sexes, these differences are to general to generalize them to the entirety of mankind or womankind.
Thus a meritocracy would be a greater improvement than matriarchy or patriarchy. Merits, contrary to its opponent authorities, are not based on generalizing, thought-limiting stereotypes. A man can be as skillful as a woman and vice versa. Meritocracy, after all, puts the emphasis on the specifications of the individual; specifications which are more detailed and more appropriate than sex-specific stigmas.
This is one of the reasons why I don’t describe myself as a feminist even though I support equality between the sexes. It seems, at least to me, that feminism has gone from supporting equality to condemning men for being masculine; a condemnation which by itself is sexist; the same sexism the original feminism was built to oppose.