Androgen Insensitivity Syndrome - man or a woman?
We have the wrong presupposition about chromosomal sex. XY chromosomal makeup is considered male not by any intrinsic condition of this combination, but because the Y chromosome triggers embryological processes that define male morphology: as a consequence of having a Y chromosome, you develop testicles, and as a consequence of their function, male morphology - internal reproductive organs, external reproductive organs, secondary sexual characteristics, production of male gametes and male psychological development- is achieved. It is a little more controversial, but I guess the same would apply to gonadal sex. In sum: we call XY the male chromosomal combination because morphologically male humans have, in general, those chromosomes. We call XX the female combination because humans of female morphology happen to have those. In birds, there is even a switcheroo involved for some evodevo reasons I forget.
In this case, the person with AIS has female morphology, and structures that have not developed normally: their mullerian duct derived organs. Their other sexual characteristics are female. Their gonads are "male", but we only call them that because testes happen to induce male differentiation. In her case, they did not do that because of the receptor being bugged.
As to modern biology, it accords with the ancient understanding - people with complete androgen insensitivity have no male sex characteristics. They are in no relevant sense men. The idea that gender is reducible to chromosomes is absurd, a variety of intersex conditions make that extremely clear, and there's really no scientific basis to doubt the obvious social conclusion here. Locating the "essential maleness" in the chromosomes rather than in the body is, as far as I can tell, philosophically and biologically indefensible.
Genital sex (which I am calling here more broadly as morphological sex, considering secondary sexual characteristics and sexual related development) appears to be the more obvious definition of sex.
"Genes" are not like a nutrition label, explaining what the contents are. They're a set of instructions, like a blueprint.
If somebody has a blueprint for a bungalow, and the builders got mixed up and built a ranch house instead, what have you got? You've got a ranch, obviously.