Lots of gendered rules differentiate based on sex. Once upon a time, there was the rule "No women lawyers." I think it's pretty clear that there is no physical basis for this rule.

Or consider the Old Testament rules on cleanliness and menstruation. Cleanliness didn't mean anything in relation to decreased frequency of disease - it was a requirement of social isolation of women from men at certain times of the month. Although menstruation is a physical thing, I would assert that these rules were gender constructs, not sex constructs.

Likewise, I assert that the rules about which gender can use which bathroom are gender constructs that use sex to differentiate. If the male and female genders were constructed differently, the Italian senator you mentioned wouldn't have felt justified in making the complaint she made.