Unrelated, of anyone in medicine, I think the only requirement I have of them is to believe in and uphold Hippocratic Oath and not let their personal beliefs interfere with performing their job or the oath. This example used, that of a person you would as a patient demand trust from more than a scientist (like an astronomer) is questionable fear mongering.
>I think the only requirement I have of them is to believe in and uphold Hippocratic Oath
Have you read it?
[In translation] "I swear by Apollo, the healer, Asclepius, Hygieia, and Panacea, and I take to witness all the gods, all the goddesses, to keep according to my ability and my judgment, the following Oath and agreement: "
Also no injected drugs to cause death or abortion might be a bar to some doctors.