Where I live, they sure do. I have a quarterly blood check, and many of my lab techs over the years have been male. I’ve seen male nurses attend my son in the hospital, male aides start IVs, etc. And of course plenty of female doctors. Nobody seems to have expectations either way about the sex of their healthcare practitioners, at least not in my awareness.
One of our formerly active members here liked to joke about his Transylvanian ancestry and the fact that he worked as a phlebotomist—on the night shift.