Gender exists as a social construct, the beginning of your question shows that you know that. So yes, gender exists and the patriarchy exists? No one is claiming gender doesn't exist in the U.S.
I don't have much of an opinion, but I'd still like to point out a flaw in your reasoning.
"gender is a social construct" does not mean "gender does not exist." Those are two different things. In fact, if a "social construct" does exist, then so does gender.
If gender does exist, then nothing is yet implying that male and female don't exist.