You say "how snowflake this country has become" as if this argument hasn't reared its head every Christmas for twenty years. This isn't a new debate and it never has been.
My two-pence, for what it's worth: I think it's important to accept that art from a previous era won't always meet the social standards of modern society, and asking for a song to be banned is unreasonable. One line in the second verse of 'Fairytale of New York' containing gendered and homophobic slurs doesn't mean the song is evil or terrible or doesn't deserve to be played. It's a beautiful, tragic song that deserves its place as a timeless classic that's wheeled out on an annual basis and a couple of outdated terms don't justify wiping it from existence.
But the problem in this debate has always been some folks' inability to understand the difference between censoring something and outright banning it. Censoring two words in the second verse of a song when it's played on radios and in shopping malls because they're no longer suitable to be publicly broadcast is a reasonable request and isn't akin to enforcing groupthink. If you want to listen to the full, uncensored version of the song then purchase a Christmas CD and blast it at full volume at home/in your car. Nobody can stop you and nobody should.
This isn't an attack on language, it's just a part of the debate some people should probably consider before they drunkenly shout homophobic slurs at the office Christmas party to impress their drunk mates. Nobody wants the song banned, they just want less than 1% of it muted when it's played in public. There really is nothing else to it.