The way I see it, Odium isn't actually taking pain, he's providing a way to shift blame, if that makes sense. To use Dalinar as an example, Odium tells him that everything that happened was Odium's doing, or the Thrill, but whatever it was, it wasn't Dalinar's fault, trying to create a sort of emotional distance from the events that, for lack of a better term, would create emotional calluses. Now if Dalinar had agreed to let Odium take his pain, it's essentially blame shifting, and as I see it, leads to mindless following, 'it's not my fault, I'm just following orders', that kind of thing.