I personally believe in the theory that the fact that none of the examples in "Ironic" are actually ironic is what makes it all ironic. So despite what the OP implies, Alanis Morissette has actually pulled a good one over most listeners.
Side note: One of the exchanges I remember most from my high school English classes was when our teacher taught us the meaning of irony and someone asked, "Is sarcasm irony?". My teacher paused and said, "Kind of, but it's the lowest form of irony". The more I think about it, the more apt that description seems.