Haven't RTFT, so this may already have been addressed, but...
Embryos do not randomly accrue characteristics. As Dr Emma Hilton memorably said, sex development is not like a Mr Potato-head. Sex is the organisation of the body around the production of either large or small gametes and sexual development is a pathway which is very occasionally disrupted or stalled. DSDs have specific causes and those factors affect either normal male or normal female development.
Individuals with CAIS are male. They have no internal female organs, they do have internal testes and produce testosterone in the male range. They are seen as women because their inability to use testosterone means that it is synthesised into oestrogen, which they can use and which results in an apparently female external phenotype. So what you see is a woman. Except that their body is not organised around the production of large gametes, which is what "female" means. Socially speaking, there isn't any point differentiating between CAIS individuals and women, but it matters for sport.
CAIS has a genetic cause and can be part of the genetic makeup of female embryos as well. Being totally unable to respond to testosterone is just as abnormal for females as for males. However, the sexual development of female embryos is not affected, because it is female sexual development and therefore not dependent on testosterone and other androgens synthesised from it.
So CAIS only results in DSD in males.
An individual with Swyers could be regarded as male in the most technical sense possible - the big switch which starts male sexual development fails and female development is therefore not suppressed. In Swyers, ovotesticular tissue remains undifferentiated and therefore produces no gametes. Only small amounts of both female and male sex hormones are produced by other glands in the body. Swyers individuals are regarded as women because it would be ridiculous not to do so given their almost complete female sexual development and indeed they can sometimes bear children with a donated egg and the support of exogenous female hormones. Like with CAIS individuals, society subjects them to the same socialisation as women.
Finally, nothing you say earlier about sex being multi-dimensional, etc, contradicts the essential facts. Sexual reproduction is a biological process based on the fusion of two gametes - one large, one small. Half of humanity contributes large gametes (assuming normal development) and the other half contributes small gametes (assuming normal development). Two sexes. There is no other way of making new humans than by fusing the genetic material of a sperm and an egg.
All of this is biological and nothing to do with subjective feelings.
Even if there is a biological cause for why a person feels they should be the other sex, the fact remains that this has no effect on what sex they actually are.
ETA: Inevitably, I have cross posted with loads of pps!