A hate crime has to involve hate. There was no hate in Fairy Tale of New York. No crime is committed by playing it, so we're not talking hate there.
Then there are words that are no longer socially acceptable. The n word is one. ***got and other of the gay abuse words are others, if used now. But in artistic work - like a song - in the past? I accept it's a fine line. Personally, I wouldn't change it but others may take another view. It's not, and should not be, a crime. Nor should using bum bandits etc as gay terms be a crime, unless used with hate behind it.
Of course, it is up to publishers, internet platforms, football clubs etc to ban someone who uses those terms, even if not used with hate. And then there are out of date philosophies like "a woman's place is in the home", "men are men and women are women" etc. Dinosaur views you may say, but it gets dangerous when steps are taken on a contractual basis to bar people from expressing those views. Like no platforming. Or sacking people because of non-pc views. That is what Rowen Atkinson was getting at. He was not referring to hate crimes.