Analogies are tools to assist understanding, and having opposition debate the analogies themselves instead of the actual points they’re used to make is a sign of a weak rebuttal.

So let’s ignore all the haggling over the analogy and bring it back to the broader point: People should not be in jobs which their personal beliefs prevent doing significant or important aspects of. And equality between genders is objectively an important aspect of health care. These “professionals” should not be in the health care field at all, save perhaps male-focused care fields like prostate or testicular health.