And barring people from "labeling things" is censorship, so I don't understand your point.

Speech that is hateful towards people because of their identities is hate speech. That is the widely accepted definition. It is not even a legal designation, nor does it need to be to concretely exist.

Denying the existence of hate speech isn't logical, because there are many unambiguous examples of it, and individuals and businesses have plenty of valid reasons not to tolerate it from the company they keep.