I suspect Masako has the point here. When people sing it in party mood they don't care if they are accurate or not, they are out to have a good time. It's only pedants who disapprove, but since no-one really knows the words of the song so what?

I remember a previous New Year's Eve programme when Richard Digance sang the whole thing correctly and confused the Scots audience who wanted to sing 'for the sake of..'

I would still appreciate a response from anyone who has ideas or knowledge of the source of the departure. I think I can remember singing 'for the sake of' at least 30'ish years ago, but memories of such times generally fade into alcoholic inaccuracy!