More generally, what Arrow showed is that group choice (aggregation) is not like individual choice...
Arrow showed that when a group chooses, there are no underlying preferences to uncover--not even in theory.
I think that most people who have had Arrow's theorem explained to them once remember it as saying something about voting. Many others would remember it as saying something like "You can't have a social welfare function," which means that you cannot get from individual preferences to group choices in a way that satisfies a particular set of axioms. Alex is saying that, conversely, you cannot get from group choices to "group preferences" that satisfy those axioms.
I think Alex comes the closest to dealing with my issue, which is that I would like to know the practical import of the theorem.
Suppose that one poses the problem as "Create an optimal decision-making algorithm for resolving group differences." For any individual, the algorithm is optimal only if he always gets what he wants. Obviously, no algorithm is going to seem optimal to everyone. However, the "nondictatorship principle" says to me that the algorithm must seem optimal to no one. Maybe I'm being dense, but I do not see why that is important.
To put this another way, suppose that this were a purely an internal problem, posed to your "inner economist." That is, you have multiple selves--a romantic self, a calculating self, a spiritual self, and perhaps others. These selves have different preference orderings. Does the Arrow theorem say something about how you can or cannot decide what to do in a situation where the selves disagree? Or does it simply say that one is going to be ruled by, say, one's heart?