@freemo I think I’ve spotted the confusion here. All the terms under the LGBTQ+ umbrella are mostly about personal identity, so what people call themselves and want to be called. This is the context in which “queer” very much makes sense, either for people who haven’t yet figured out the details, but know they don’t fit the cisheteronormative default, or for those who did not end up fitting into any of the other boxes fully (plus some political meaning, but that’s kinda separate).
If you want to be specific and precise when referring to a group then there are almost always better terms – at least in medical, law, and social contexts, I cannot think of any other relevant ones. The specific division you advocate for here is extremely rarely appropriate anyway – you almost always want to refer to a strict subgroup of one of the groups you described, or to a group that encompasses people from both groups.
Having said that, while I can argue that the division you propose is bad on purely practical grounds, I have to also point out that categorizing humans has an extremely fraught history to say it lightly, so ignoring the political implications of any proposed categorization is, in my opinion, extremely unwise.
You seem to be somewhat confused about the gender/sex distinction. In this context “sex” does not refer to just genes, but general biology, including hormones and the phenotype, while gender expression (I have to specify the second part here, since “gender” can, confusingly, refer to at least two other concepts that are very relevant to the discussion, but fortunately not to the distinction here) refers to social indications of gender (behaviour, dress, etc). Thus being intersex is purely about sex, regardless of the specific syndrome. Some people have argued this is a reason why they shouldn’t be included under the LGBTQ+ term at all, but it turns out that their interests politically align with the group often enough that they usually are.