“There’s a river of love that runs through all times,” the singer-producer T Bone Burnett sang in a signature song of his back in the 1980s. But that’s not a river he’s necessarily always been riding.

A T Bone Burnett who is embracing the spotlight as an artist would seem to be a different guy than the fellow who’s often insisted over the years he prefers to stay out of it. But, as he tells Variety ...