No it’s not and there’s scientific evidence. Studies have shown that female children will name career paths as their dream job less often if only the male version is shown to them, presumably because they think it’s not an option for them, as it’s perceived to be only possible for males to follow that path. Explicitly mentioning both genders suddenly makes girls also want to become doctors.

You might not like it, but there’s enough evidence to show that it has merit.

I mean there are no formal rules on how to form a gender neutral noun yet afaik, but what you stated is the form that would be used most commonly. Ärzte’innen is definitely nothing that I’ve seen so far, especially with the ’ character. Usually * or : is used

German even has a neutral gender because two aren’t enough of a headache lol