None of this is relevant to volver's question; but my understanding was that JKR decided to publish the Strike books as Robert Galbraith as she wanted to see if she could succeed as a detective writer without the halo effect of her success with Harry Potter.
I'm not sure how that was going to work, as it very soon became common knowledge that it was her pen name, but I don't think it matters. Several authors write under more than one name, and it is ironic that it is those who are most vociferous about sex and gender being changeable who are objecting to her choosing a male name.
As for carers, doctors or anyone else being male, of course that is fine. If a woman would prefer to be treated or cared for by a female, she can ask, and male staff carrying out intimate exams usually have a chaperone. It is when a man says that he is a woman that it could pose a problem. Does anyone know whether in that case there is a requirement for a chaperone to be present when a female patient is being intimately examined?