As you say, fairy tales are elemental and stark. Primal, one could say. I think Disney has conditioned us to think of fairy tales as stories for young children, but most of them are adult folktales, or were written for adults originally. They're also, as Rachel points out in her comment, very woven into our culture, for better or worse.
I tend to subscribe to the general sense of Bruno Bettelheim's analysis of children and fairy tales, summarized well in this article. The situations and characters in fairy tales, Bettelheim says, reflect thoughts that children have, but don't know how to process.
Bettelheim argues that what children really gain from these grim and troubling stories of being shoved in a witches oven, having one’s head cut off, or being eaten by a wolf (all not exactly a ball of fun) is the symbolic assurance that all difficulties in life can be mastered with the right attitude and some courage. Folk tale characters experience existential anxieties that are inherent in human nature: the need to be loved, the fear that one is thought worthless (look at poor Jack with his handful of beans!), the value of life and joy, and the ever-present spectre of death.I've spoken with a number of teachers who read fairy tales to kids in grade school, older than the ages that parents usually turn to these tales. I often recommend fairy tale collections as gifts for kids who are five, six or seven. Even though children those ages are able to pay attention to much longer books, the emotional intensity of fairy tales, and the leaps of imagination they entail, appeal to children who are dealing more with how they fit in the world. They can envision themselves as grappling independently with the world, even if their reality is quite sheltered.
I've never been a fan of the young adult horror genre, but it has a devoted following. I think the motivations for reading it fit into Bettelheim's analysis: facing one's primal fears, and vicariously vanquishing them. Again and again. One gains strength by seeing what it's possible to overcome.
Adam Gopnik wrote an interesting piece on fairy tales in The New Yorker back in 2002 which rambles a bit but discusses both the Bettelheim take on folktales, and a more conservative they-teach-you-good-conservative-values argument.
Such a rich vein, fairy tales.