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.
1. It's not Critical Gender Theory, though Critical Gender Theory uses broadly similar definitions for the subset of those things which it thinks are worthy of discussion, it's just the fairly common language to distinguish the innate physical (sex broadly, reproductive function for the particular subset you raised), internal self image (gender identity), externally ascribed status (gender), and behavior (gender presentation), and neutral as to material as opposed to conceptual relationships. CGT is opinionated about the material relationships, holding (as I understand it, but CGT isn't something I've more than casually encountered) that gender presentation is socially learned behavior produced by social gender ideas and other aspects of personal social context, gender identity (which CGT doesn’t, IIRC, talk much about directly, but recognizes insofar as it recognizes that people have a social identity related to the social construct of gender) as a product of gender presentation and social gender stereotypes (reversing a common progressive view that gender identity is a fundamental, immutable or nearly so, aspect of identity amd that gender presentation is an expression of identity, to which social gender images may play a part.)
2. It (my post, not Critical Gender Theory) is not supposed to be scientific; it's pretty much all definition of terms, which precedes the ability to even discuss observations of facts, which in turn precedes the ability to discuss hypothesized generalizations from observed facts and empirical tests of those generalizations.
It's important to remember that essentially all human concepts are socially constructed on some level. Even basic ones like 'fish' (is a dolphin a fish? Many cultures would say yes, but modern taxonomy classifies them differently), 'fruit' (is a tomato a fruit?), 'river' (is this a river or a canal?), so it would be truly remarkable to find out that the concepts 'male' and 'female' correspond to objective physical reality - they would be unique among human words!
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.
