That's a decent, if slightly confused, defense/description of the view of gender and sex prescribed by Critical Gender Theory, however, despite its academic sounding terminology and many PhDs, critical gender theory remains unscientific hokem.
So no, you don't require CGT to make a distinction between biological sex and the fuzzy notions of 'woman' and 'man'.
To give some examples: is a person with XY chromosomes but Androgen insensitivity (testicles, but a purely female phenotype, including all anatomy, such as a uterus) Male or Female? Purely biologically, obviously male (small, mobile gametes). Would you send this person to male prison? Would they be able to inherit the title of King in medieval society? Should they be encouraged to dress as a man, because that is their biological sex? Should they not be allowed to a gynecologist, since gynecologists are only for women?
Similarly, let's say you have a person with XX chromosomes born with a typical female phenotype, but who has undergone bottom surgery and testosterone therapy, so they now have no breasts, hair on their chest, a penis and apparent testicles. Is this person female? Biologically, yes - they have (or had) large, immobile gametes. Would women feel comfortable if such a person were to undress in a female locker room? Should an advertising company seek to target tampon ads to this person, since they are biologically female? Should they be allowed as a contestant in a Miss America pageant?
The discussion of how malleable the relationship between sex and gender is, and CGT does make arguments in that area that are arguable. But it is not arguable in any way that the concepts 'man' and 'woman' are much fuzzier than biological sex, that they are socially constructed, like all other natural language words and concepts.
As to classifying what dragonwriter has written as hokem people would find it more convening if you gave an actual argument.