Why are you using police (who have no legal or ethical duty to care for or protect anyone) as a baseline for what we would expect of healthcare providers (who have both a legal and ethical duty to their patients)?
AFAIK. it hasn't been copied in that form very widely, instead, it was popularized by TV (Dragnet specifically, IIRC), and so is culturally attached to police in general, despite the vast majority of departments with any motto having different mottos.
That is a PR slogan. That's like trying to say Google can't be evil because their motto is "Don't be evil."
The Supreme Court has explicitly ruled that police do not have a duty to protect citizens, and that ruling has been reinforced consistently at lower courts throughout the country.
No, that's not what they meant. The ruling itself is extremely explicit. And furthermore, that's how other courts have applied the ruling in the decades since.
It's simply factually incorrect to claim that the police have a legal duty to protect citizens simply because of a PR campaign that they launched, when there are volumes of case law that establish otherwise.
They do have a responsibility to protect people in their custody.