I don't see why self-identification for legal and institutional purposes is in any sense strange: we use it for lots of things related to legally protected characteristics, and it's frequently by far the most sensible way of handling a range of issues. Any attempt to create neat, medicalised categories in this area tends in practice to exclude and harm people who don't fit those boxes, not to mention often being deeply invasive. I think, too, that it's not quite as simple as "respect but disagree": I'd tend to consider it an important feature of respect to believe people when they tell me a fundamental characteristic of who they are.
In any case, I think your case above is also predicated on an incorrect assertion about sex, namely that it's less complicated than gender. It's possibly more complicated, because "sex" is a way of sorting humans into categories (which do not form a neat binary) based on a vast, vast range of different characteristics, many of them mutable, including chromosomes, a wide range of secondary sexual characteristics, and hormone levels among other things. There's no way to test for, or perfectly categorise, sex, any more than there is for gender. People whose sex is female for an awful lot of medical processes include, for example, trans women taking oestrogen, because that's, well, kind of the point of oestrogen. The fact those people still have an XY chromosome structure isn't a sufficient bit of information to sort for medical processes. The implications of trying to check and test people's sex for their general interactions with institutions are deeply troubling to me due to the arbitrariness with which some of these categorisations would end up being made and due to the personal invasiveness of checking people's medical details in various ways for access to services. Where there are specific medical things that rely on certain body parts, hormone levels, and so on, we are scientifically at a point where we can (and should) be more precise about who things will and will not benefit than trying to lump the population into a strict binary when actually the edges are very woolly indeed.
Regarding what gender is: yes, it's a social construct through which we understand a lot of these sorts of sex and performed characteristics, and yes, it can be categorised differently across cultures. It's clearly true that societies do tend to exert pressure towards gender conformity - but it's also true that some of the characteristics we understand through gender are fairly innate, precisely because gender and sex are different layers of social understanding of our bodies and societal roles that can't be fully disentangled. As such even if you remove pressures for a gender to act in a certain way you should still expect that people will not all necessarily be OK in the gender and with the sex characteristics that they were assigned at birth. Indeed, the expectation that people must retain their birth gender contributes in the main to societal pressures towards gender conformity. Excluding trans women from womanhood starts producing a list of "people with these characteristics aren't really women" - and you invariably find that a lot of cisgender women share characteristics with some trans women, especially when those are things like "some women are taller than average and have deeper voices and a more square jawline" (which are the sort of cues people practically use to assess gender or sex in public spaces).
For me, what I care about most in this are the practical questions like "can people I care about go to the bathroom without being harassed for not conforming to someone else's idea of what a woman should be", and "could trans women I know use gender-appropriate domestic violence shelters", and "can medical specialists get on with treating trans children for gender dysphoria without being defunded due to an arbitrary moral panic". The answer - which I believe to be extremely harmful - of the "sex is inherent and chromosomal" lobby has invariably in recent years been that children should not get affirming treatment for gender dysphoria, that trans women should not be housed in women's prisons or domestic violence shelters, and that trans women should not be able to use women's bathrooms. Those things are not supported by the shelters or medical practitioners in question, or indeed women in general, and have led to extreme levels of personal harassment to people I care about. That's the practical, day to day reality of what this particular thought trend is currently doing and looking like in the UK. The abstract theory points are something I'd be much more interested in discussing if it wasn't for them being used as the backdrop for a reprehensible political agenda, the primary platform of which advocates for things that will cause genuine suffering to trans people. I appreciate that most people don't really get to see just how under siege trans people in the UK feel right now or some of the hatred that gets directed against them, but it honestly scares me.