I voted YABU because people are looking at the case in point all wrong. The doctor is allowed to identify as female, therefore the doctor is providing same sex care. The problem isn't really the doctor, the problem is the system that allows a man to become a woman and a woman to become a man.
I have said this before and will keep saying it. As a society we need to decide once and for all whether someone can change gender. It's a straightforward "yes/no" decision. Let's have a referendum.
If a man can become a woman, they should be treated as a woman in all circumstances - they can compete in women's sports, go to women's prisons, and in the OP's example provide same sex care.
If a man can't become a woman, they must be treated as a man in all circumstances regardless of what they claim to be and what they believe themselves to be.
The problems we have at the moment is because there is a middle ground. A man can become a woman sometimes and in some circumstances. A lot of people say "I don't care how a man dresses or what pronouns they use, but they shouldn't be in our changing rooms." Fine - but ultimately you're saying a man can't become a woman. If they can't become a woman, them dressing in female clothes and calling themselves she/her is nonsensical and offensive, the same as someone blacking up and casually referring to friends as the n-word.
Let's have a referendum and settle it once and for all. (Or at least once and for a few decades.)