My answer is yes. Gender is a social construction created, however, in response to our biology of sexual reproduction. The function of gender is to prescribe the social conduct required for sexual interactions. While biology determines the sexual apparatus of the body and embeds in us procreative instincts, gender is a set of forms and behaviors that we adopt being strongly induced to, inducted into indeed, by our early environment and mostly by parents who more or less apply the norms prescribed by society for a biological boy or girl.
Gender becomes strongly embedded in the body as it relates to the sexual aspirations and wishes. That means gender is about how we will realize our sexuality, about how we will couple, not just relate, with others. Just because it is a social construct does not mean that it is easy or painless to change it or go against it. Therefore it is not purely performative as a social role.
Gender is the way we present our sexual aspirations to ourselves. Such presentation in principle should admit a huge variability. This option is eagerly embraced by the proponents of a "gender spectrum" or continuity denying the validity of masculine and feminine gender expressions. For them the cultural standard should be really a choice of the proper ratio of the masculine and feminine.
So why would we have two genders? Just because there are two biological sexes?
Partly yes, but since we are speaking beings that interpret everything and try to apply and extract meaning to and from each and every fact and act of our lives, the gender expression is due to the semantics of sex rather than to its reality. In other words, because we speak we express sex through gender - that is, we make it complicated. Alenka Zupančič takes it further: because we have sex, she says, we need to develop language to handle it.
In the 1990s, Joan Copjec, a feminist, wrote an article deriving gender difference from the antinomies of Kant. According to her paper, man is the dynamic failure, meaning that he does not rise to the occasion, while woman is an existential (or mathematical) failure, meaning that she does not exist. The latter statement is, of course, scandalous, and of Lacanian provenience.
Let me recount Copjec's argument in the form of anecdote.
There we have It - the elusive and obscure signifier of desire. Hidden in the unconscious and totally inaccessible. Lacan calls it the Phallus.
The man has It yet knows he is not It but because he has It he is sure he will become It.
The woman is sure she is It yet she does not have It but is sure that somehow she will obtain It.
This is really a Lacanian version of Freudian "penis envy" - now distributed between the genders, albeit asymmetrically, owing to the split between the modalities of having and being.
In explicit and yet very elucidating sexual terms we can say this:
The man has it (the penis) but does not have the erection. So he is not It.
The woman is It (she is the erection) - but she does not have it.
Now the failure, the lack, is distributed - not equally but so that each gender can complete the other. Just like it was explained by Aristophanes in old Plato's account.
Speaking more closely to Copjec's terminology, man's dynamic failure is the failure to master the delivery of power - as he presumes to have what it takes but is challenged to make it work, cannot make it It. Woman is It but does not believe it - she has to have somebody prove it to her. Man can do it as he directs the power he creates toward her. The proof is seen as love. Love proves that she is It.
Casting this discussion in terms of power - and I consider power a transmission of energy or force in a semantically organized manner - seems to me a more socially acceptable simplification. That is why I sometimes say that gender difference is founded on the relationship to power. Man is the creator and generator of power while the woman is its recipient and beneficiary - and thus its judge.
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References:
Copjec: "Sex and the Euthanasia of Reason" - in "Supposing the Subject" - Verso 1994
Zupančič: "What is sex?" - MIT 2017. page 43
