Have you got a source for those non Kilinefelter XXY? Noodling round Wikipedia doesn't seem to substantiate, although there is a nice table of the karyotypes which is more extensive than you list.
Anyway, while I recognise it's coming from the right place, I remain sceptical of the argument that says "we have the way hell than just two biological sexes". As best I can tell, we do have two (maybe three) sexes: one with testes, one with ovaries and perhaps one with ovatestes. That decision is driven by the karyotype (generally a functioning SRY gene). And if everything works through to puberty, then the gonads will deliver one of two phenotypes (with it being unpredictable which phenotype someone with ovatestes will end up with). Most people fall clearly into the phenotype associated with their gonads, but a few don't. Some people, for whatever reason, end up feeling they are the wrong phenotype. Those people would do anything to be be a "normal" man or woman. Multiplying the number of sexes doesn't help. You're saying "hey these people might be in another category entirely"; it's more or less like a cis woman who says to a non-cis woman, "you don't have my experience, you have your own experience". The argument can't be won by pointing at a gene or brain scan; the phenotype people feel is a subjective thing and that has to be accepted, before we even get to the performance of "gender".