I haven't read The Selfish Gene, but from what I've heard it tells the exact opposite: That every gene's goal is to propagate, for which task it merely employs the living organism.

Talking about "goals" and "desires" can sometimes be a usefully simplified way to think about biology, but it's not really accurate. A bacterium doesn't "want" to reproduce anymore than a rock "wants" to roll downhill. Only quite complex animals can be truly said to have goals.