For what it's worth, I'm in a production of Sweeney playing Turpin. We staged it delicately, with all of the the self flagellation and orgasm. I believe the song is important as it informs the audience about the tortured pathos in the judge. By doing the song, and handling his dialogue differently, as an actor you have the opportunity to show the true hypocrisy of the character. It also helps explain Sweeney's being driven to mass murder in an attempt to kill his nemisis.
A few days before opening, the theatre decided to cut the song from the show because they felt it slowed the momentum of the storyline. I felt like the judge had been defanged. Without the song, he's another moustache twirling bad guy. I mean, yes, the show needs a bad guy, but the character has nowhere near the depth it did. I have found a few ways to take the essence of the song and infuse it into the dialogue, but it's not nearly as layered.
The song should stay.