whether you are male or female is a matter of fact / of birth / of biology.
It is obvious through appearance
It is as much a part of someone's biological makeup as the colour of their eyes
It is not personal medical information.
It is not an intimate detail
Hope that helps clarify your confusion.
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There is no such thing as a 'real vagina' as that implies that there is such a thing as a constructed or other vagina...
there is just a vagina. - surgical butchery ending with holes in someone's body - does not make a vagina.
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If someone is trying to pretend that a man is a woman (when we know that is not possible), then that is a deceit, it is not factually correct, so we have to ask why they are doing that - and there are only two apparent reasons...
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I don't need to believe that sex is immutable - it just is immutable. Much like I don't have to believe that the earth is round, but that doesn't stop it being a sphere and make it flat!
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an individual doesn't get to decide when it is relevant for a HCP to need to know that they are e.g. male and not female - it is a necessary piece of medical knowledge which can affect virtually every medical decision. The NHS is not yours to dictate how it acts - it acts in a manner which is proven to be best for the patient. Even something as innocuous as dealing with a minor injury might need that knowledge to make decisions on pain relief and other treatment...
As for a receptionist knowing - so what, their job means that they need to know all sorts of information about patients - that doesn't mean that they will use that knowledge to the detriment of the patient - so the concept that up and down the country GP practices are refusing access to medical care because a receptionist knows the sex of the patient is quite frankly absurd.
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We are part of a society that has a structure which makes it work. The sex of an individual is a core part of that - we have everything from employment law to toilets, education decisions to medical decisions, even marriage decisions (e.g. Church of England v. registry office) all which come down to knowing the sex of an individual - it is so core a part of who they are that it usually has more relevance than the person's name. Sorry if you don't like it - but one small group's dislikes doesn't set the pattern for our society.