...I'd file it under Hypnotic Masterpiece. Just like every Aronofsky film I've seen it roughly a dozen times and it keeps getting better.
For me, the story doesn't ask for the audience getting a deeper connection to the main character. Actually, I don't think there's a typical main character in there at all...just as there wasn't any in Pi and Requiem For A Dream; the film is all about a quest(ion), whereas the characters are possible ways to cope with it. It didn't come across to me as though the heavy emotions of loss were to be passed on to the audience, but to trigger the same question in it; hence it isn't a catharsis type of Hollywood tragedy, but one that demands the audience to step into the point of view of the characters without necessarily connecting to them.
That's what it does for me...maybe it also has to do with the fact that I don't really know how to cope with death myself, as I've never witnessed any (tragic) death cases in my social network. I'm not sure if I would see it is a natural part of life...it might well be that I'd see death as a disease as well, just as Tommy does.