Playing devil's advocate here, some automated systems insist that a 'title' is added to a person's name (e.g. Dr, Ms, Prof). If you don't select one, it can't cope so adds a generic title like Mx.
That doesn't help you, I know, but it's not necessarily obfuscation - just a computer system that has to have a title.
You can also ask to have a chaperone for any medical appointment - either a friend/relative or a member of practice staff (of your own sex, if you ask). You can also request care from same-sex practitioners, if it's available.
Fwiw - and I'd offer feedback to the practice on this. It isn't transphobic to ask that you receive care from someone of your own sex, where available. Lots and lots of people - for religious or cultural reasons - prefer this and will ask for it. There's a male nurse at my practice and both when booking appointments and at the appointment itself reception staff and he always ask if I'm okay with being treated by a male nurse (and I am).
Ask that staff name and sex is included on the website. My practice does this. It's especially helpful when staff have names that aren't obviously male or female (e.g. traditional Sikh names can be be given to boys or girls, or more modern English language names like Morgan or Taylor which both men and women can have).