Whoa. No need to bring TRUST into it. I repeated a story I heard. I wasn’t there. You weren’t there. I don’t know who said what to whom. Neither do you. I extrapolated my question, “would it bother you if you couldn’t see your doctor’s face?” from that sketchy anecdote.
A hijab shows the face. So not a problem IMHO.
As for religious garb, or clothing not like that worn by regular folks walking around in the street today: Not so long ago entire hospital systems were run and staffed by nuns who wore clearly identifiable religious garb that made them stand out in today’s world, but which, ironically, was often chosen at the time their orders were founded in centuries past, to make them blend in with other womenfolk. Often a nun’s habit was the common garb of a widow of that time and place. The idea was NOT to stand out, but as the centuries wore on, well… they really stood out, which is why many switched to white blouses and skirts from Wal-Mart, so as to blend in better.