Honestly I've always just put it down to another fault of the English language being weird. There should be a more concise word for a singular person where the gender is not defined, but there just isn't so people use "they".
It's acceptable to me if the number of people isn't known but in your case of the doctor I would not find it ideal but acceptable just because there isn't another proper word if you don't want to / can't use the word he/she.
To me in an ideal world They would strictly be for multiple people or an unknown number, and another word for those who don't identify as he/she or for cases where gender isn't relevant like title professions and whatnot.
I think we have. In my area many deer during hunting season spend the time within city limits where it's illegal to fire guns. I am pretty sure they've figured it out.