Trans language isn't clunky, it's just dishonest and sodding ridiculous ("assigned female" ffs). The language I have to use when you will insist that some men are women - so there is no inoffensive term that actually has a definable meaning, since there's no such thing as a woman if men can be women - is clunky. That's the point. We are all very used to "trans inclusive language", thanks... you may not have noticed but it's been forced on us for a while, women had to go through several courts because it was rendering everything meaningless and now it seems we will still be up against the Newspeak if we visit sexual health clinics and ask that a woman attend us.
So here's the key:
No, they wouldn't necessarily send in a person assigned female at birth
So a vulnerable woman (or not, there isn't actually a purity test for women's rights) requests a woman for her intimate examination, and you might well send a man, tell her he's a woman and tell her it's not ok if she requests "proof", which she would only do because you've lied.
Incidentally, why do you think it's relevant that it doesn't happen often? Is there an acceptable number of times? Is it actually a bad thing so you wish to mitigate it a bit?
I find it staggering that you'd do this, see nothing wrong with it - it's clearly prioritising a dishonest ideology over a patient's consent - and accuse others of being the ones to make too big a deal about transgenderism.
I would ask a lot more questions but I know you won't answer them. Can we at least know the name of the clinic that doesn't respect a patient's request for a female practitioner?