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My understanding is that if you move the binary to functional SRY gene, you have a near binary with the possible exception of androgen insensitivity. So is sex binary with the claim: a person is male if they have a functioning SRY and are androgen sensitive, else they are female?
There's probably some genetic abnormality aflicting 0.001% of the population I'm not familiar with, but this seems near enough to binary to me.
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Why is the chromosomal definition not binary? Presence of Y chromosome => male, everything else => female. To me that is binary. >

I guess it depends on who you're talking to about it. Please note, I'm not an expert in these fields, just someone who has some interest.
If you are interested in "binary sex" from a scientific perspective, Y chromosome certainly is a very important thing to understand, but ultimately it doesn't correspond 100% to binary sex, so you should understand the definition around gamete size (eg it's possible to have an XX with small gametes). There is very useful science to be studied around chromosomes. However, modern science relies on the gamete size definition. Honestly, as a lay-person, this makes sense since "sex" is meant to describe an aspect of sexual reproduction, and gametes are the key to sexual reproduction.
From a medical perspective, binary sex is only useful to a degree. Biological definitions around gamete size (or presence of Y chromosome) don't correspond well enough to the various phenotypic traits you describe, and if we try to apply a binary sex model to individuals in a medical setting, a decent percentage of people will have worse medical outcomes. Eg. it's possible for an XX to have more "male" phenotypical traits. So a nonbinary model is very important here.
If you're talking about it socially, the social idea of "sex" is very much nonbinary. Y chromosome (and also gamete size) doesn't correlate very well with the sexual characteristics that people often think of when discussing sex, with or without medical intervention.  So a nonbinary model is useful here too.
Conflating social, medical, and scientific definitions often leads to arguments that are unproductive as people talk past one another. So attempting to raise binary sex into social or medical conversations may end up being frustrating or counterproductive in some cases. >

!delta yeah I love this the social definition, and the most common one definitely isn’t binary. >

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Might i just say that biology is never binary. everybody, literaly every "body" is unique. the production of hormones that detemine your primary and secondary sexual features, the way you think and how you identify are all regulated by physical processes. for example the amount and ratio of hormones you produce, the development of specific areas of the brain, and mutations on the genes that turn off, limit, promote or turn on certain aspects of the complex chains of steps that your body takes to develop into what ever it developes into. there are so many factors that determen your biological sex, and variations there of, that it becomes difficult in some cases to round someone up or down to the nearest sex i.a male or female. sometimes a body can truely be between or beyond these two options. and how the sense of self is altered by your enviourment, genes, and culture is so personal that gender can sometimes not align with someones biological sex. 
in a black and white world there are only 2 sexes for humans. in biology there are no real sexes, there is just life and we classify it and put it in boxes. 2 boxes is the easiest to work with. but there are spiecies that have 1 sex, 2 sexes, 3 sexes, 5 sexes. there is no limit to the variation. and what definition we use to determine a sex is also made up. nature didn't write it in stone or in the skys.
as long as you're happy with how you are and can accept someone esle for how they see themselfs there is no reason for being argy about people not fitting in our self made boxes, or wanting to be places in the other box. >


Male => has Y chromosome, female => does not have Y chromosome. This definition is inclusive toward those with chromosomal differences such as those with Kleinfelters, Turners, etc.

This isn't strictly speaking entirely correct because XY gonadal dysgenesis (with Y chromosome, but dysfunctional SRY group) exists, which presents externally as female. 46,XX/46,XY chimerism also exists, which can result in either indeterminate sex characteristics or just entirely male or female presentation.