Never cared for the song myself. It doesn't annoy me (Sweet Home Alabama's response to Neil Young does), but I would defend anyone else's right to be troubled by the Lost Cause posturing. I grew up in the South during the last gasp of Lost Cause mythologizing, and I know the insidiousness of fake nostalgia for a cause that should have lost and been forgotten, and how that fake nostalgia generated and preserved racism.
So if someone wants to object if a radio station plays the song, fine. That's their right. If someone wants to play the song at home and can do so without engendering self-deception and racism in themselves, that's their right too.
Incidentally, "Dat's Why Darkies Were Born" was apparently included in a musical review as a parody of the attitude that it seems to proclaim. But later, some singers didn't get the joke--most notably Kate Smith, who belts out an earnest version. I first heard it covered satirically by two British musicians--avant garde saxophonist Lol Coxhill and classical composer David Bedford (on Coxhill's "Ear of the Beholder"). They preface the performance with a comment about how stupid the song is and how since the writers are dead, they won't get any royalties from it. Maybe one day, it won't be unusual for people will look at "The Night They Drove Old Dixie Down" the same way.