Here's a 1979 commercial for the original Broadway version of Sweeney Todd.
I always find it interesting that the show was not really advertised with the horror/blood aspects played up, the way the movie is being marketed; starting with the jokey poster (which placed a caricature of Angela Lansbury next to an old Victorian cartoon caricature with a Sweeney Todd razor in its hand), it was pretty clearly being marketed for its camp value, even though the actors were trying to avoid camping it up in the show itself, and continue to be rightly proud that they didn't give into the temptation to play the thing for camp.
This was probably the sensible way to market the show, given that it didn't really work as a straightforward horror story or as a straight drama. The flaw of Sweeney Todd as a piece of theatre is that it comes off as vaguely pointless. The original legend of Sweeney Todd has a point, the same point as any horror tale, which is to play on our paranoia about some real-life thing: it's about our fear of going to the barber and putting ourselves in the hands of someone with sharp instruments. But the more you inflate the story the more you bump up against the question of what the hell it's supposed to be about; no matter how hard it tries to sell itself as a revenge tragedy, it just isn't. (The story would be truly unsettling if it made us feel for a moment that we could descend into revenge the way the main character does; but because his actions are so over-the-top, we know for certain that we would not, in fact, do what he does.) I used to think that Hal Prince's decision to overload the show with ersatz "social commentary" on the Industrial Revolution was taking things too far; now I think he might have just been giving the audience something to hold onto; if they started to ask why this story was being told, the show would turn around and reassure them that it was something to do with how bad the Victorian era was.
It's been suggested, plausibly, that Sweeney is a sympathetic character until he snaps and starts killing people randomly; everything he does in the first act is sort of justified and the audience might actually be on his side. (That's the way Len Cariou apparently played it.) Given that the show appeared at the height of vigilante-justice fantasies, Death Wish movies and the like, it could work by drawing us into sympathizing with a vigilante and then reacting in horror when we see what we're sympathizing with. But that's not really an option because the show telegraphs from the very first moment, and throughout the first act, that the lead character is going to go bonkers and start chopping up everyone in sight. So a Bonnie and Clyde-style approach (first half, make us like the killers, second half, make us realize that maybe we shouldn't go around liking killers) isn't an option.
Also, it points up a structural problem with the show, which is that not a whole lot happens until the end of the first act, which makes it consist of a very padded first act and a second act that tries to get too much done too quickly. The movie makes some good cuts as well as some less-acceptable cuts in the score, but at least the cuts go some way toward counteracting the biggest problem in any Sondheim musical: songs that take five minutes to say what could be said in half the time. (Or, in the case of the Judge's song that was cut from the original production but recorded for the album, take five minutes to say what we already knew from two lines of dialogue.)
The movie, I think, is playing it more as a dark horror story that moves too fast for us to notice that the story doesn't have much of a point. Tim Burton is good at that, since several of his other movies fall into that category (Mars Attacks is one of the most pointless movies ever made, and I like it). I'll have more on the movie later this week, but I think it was quite well-done and does a good job of not making you question why this story is being told, a key job for any production of this material.
By the way, three years after 1979, Little Shop of Horrors would sort of take this marketing idea -- a combination of violence and camp in a musical format -- and run with it by sustaining that tone of camped-up gruesomeness for an entire evening.