Sex is in fact socially constructed (video source on this posts’ claims);

Sex is a biological fact, unless they've redefined the word again in the last five minutes. Did you mean gender? Also, that video is crap.

it’s a categorization scheme for biological characteristics instead of cultural ones.

Yes! Try and explain that to some people.

Sex therefore has some social aspects.

No, the categorization scheme has some social aspects. Whatever particular definition of "sex" you are using at the moment is a biological fact. There might be different categorization schemes used to define what we mean by the word "sex", but which ever one we use will be based on biology. Unless they've redefined the word again in the last five minutes, of course.