we live in the 21st century
Not sure why that means we have to ignore all potentially embarrassing/uncomfortable feelings around intimate situations with members of the opposite sex, let alone actual safeguarding.
I told myself just that when I was surprised to find a male masseur arrive for my appointment. I dare say I should have expected the possibility as I had booked in on a whim for a "sports" massage while my boyfriend of the time used the tennis court at a hotel In the sports area. Lots of sports physios are men. I should have gone to the beauty parlour instead. I told myself not to be silly. Anyway suffice to say my groin and inner thighs were thoroughly groped massaged until I told him to do my back as I had shoulder pain and he took the hint. I probably should have complained but didn't want to make a fuss. I wouldn't say never - it depends on the person/situation - I have had a male osteopath since then for instance. But in general, no, I would not have a man in that sort of situation in the future.
Of course there are lots of wonderful male Drs, nurses, HCA's, and others working in health services. There are undoubtedly some crap female ones, I know some female obstetricians who wouldn't dream of giving birth any way but an elective LSCS and who are distinctly lacking in the empathy area.
Most men working in the health service aren't doing so for the opportunities for abuse that it presents.
But there have been some appalling cases, the Addenbrooke’s paediatric cancer specialist for example.
I think when these cases present and hospitals review "lessons learned" and safeguarding and so on the main lesson learned it to be alert and don't assume anyone is above suspicion. Or something along those lines. So if some women prefer to protect themselves by choosing female staff that should be a perfectly acceptable preference.
That said, some posters talk of always choosing women, and I'm not sure it would be possible to never have a male Dr or nurse even outside of emergency situations. At night in most hospitals the number of junior doctors on duty is very small with one Dr covering a lot of areas. There simply wouldn't be the option of asking for someone else. Ditto for the number of nursing staff on duty. Or you may need to seek out the best specialist in a niche area and that specialist is male. But that is all very different to choosing/accepting a male midwife, within the UK expectation of community visits and postnatal care as well as the actual birth.