As Marje said,
"The older (Burns) tune is never going to replace the more familiar one at midnight on Hogmanay, but if you want people to listen to the words, there's a lot to be said for introducing a different tune. The Burns tune is lovely and more reflective, fitting the sentiments of the song."

And I do agree that if I tried to use the more familiar ALS tune for a New Year's party most people would balk at the idea. However I don't think even we Yanks would bat an eye if I correctly pronounced "syne" (with an "s").

There are plenty of words borrowed from one language to another culture which then are so commonly mispronounced that the new pronunciation becomes "correct" for that new culture. Very few French-borrowed "cognates" are pronounced by English-speaking people the way the French would, for example.   Still, if I'm singing a song that has French words in it, I try to pronounce them the French way.

On the subject of the pronunciation of "Auld lang syne," though, there's something I've puzzled about for years.   In an old American movie -- I believe it was the Jane Froman story, "With A Song In My Heart" -- the song "Auld Lang Syne" is sung but the singer (if that's the movie, it's actually Jane Froman dubbing her own voice) sings "Auld ang syne."
You can't hear the "l" in "lang."
Does anyone know why?
Is there some regional dialect that would pronounce it that way?