I suspect that many people, including Tom I think, are tacitly imagining voters as actually having opinions about all sorts of things. Perhaps some voter prefers that if three out of candidates $C_1, \ldots, C_n$ be picked, then it should be $C_1, C_2, C_3$, unless it so happens that my neighbor also likes that choice, in which case I would prefer $C_4,C_5,C_6$ just to see my neighbor suffer—but if my wife tells me that she hates $C_6$ then I'd prefer $C_4,C_5,C_7$ (at least, if she tells me this six or more days before my dad tells me that he hates $C_7$, and I don't think my dad is lying, and $C_7$ placed no worse than second place in the last two elections), etc. Every conceivable opinion is there in the voters' heads. Designing a voting system amounts to some method of sampling from that vast sea of information, and the designer wants to make sure that the system doesn't suffer from paradoxes.

If one is trying to generalize Arrow's theorem (and the like) in the direction that Tom seems to want to do, then I think that this way of thinking about the voters is unhelpful. The reason paradoxes arise is precisely because when the voter preferences exceed some threshold of complexity, there is no consistent way to define "the will of the people". As long as people secretly have those complex, paradoxical preferences, the result of any voting system is going to fail to express the will of the people, simply because that will does not exist. It doesn't matter whether the voting system samples more or less information or in what way; the most it can do is to make it obvious that there is no will of the people, but even if you sample information in a way that fails to lay bare this ugly secret, you have not successfully expressed the will of the people. You've just buried your head in the sand.

The way to approach the issue, in my opinion, is to drop the assumption that voters have opinions about every conceivable thing, even those that don't appear on the ballot. Instead, assume that the ballots capture everything that there is to know about a voter's preferences. If a preference does not appear on a ballot, then it doesn't exist. This is not really a limitation, because we are free to design any kind of ballot we want. The ballot could be a long questionnaire that solicits opinions about your neighbors and the last election and whether you're in a bad mood today or whatever. The only rule is that if a preference is not on the ballot then it doesn't exist. This simple convention short-circuits all the complications about whether a voting system is burying its head in the sand because it asks for partial information only, when if it asked for full information then a difficulty would be revealed. There is no difference now between what voters prefer and what the ballot indicates. This makes the problem much simpler and clearer.

Now perhaps my original suggestion will be clearer. To generalize Arrow's theorem, we want to look at ballots that do not express more preferences than a total order. It could express strictly fewer preferences (a partial order) or it could express a set of preferences that is different, perhaps seemingly more complicated, but not strictly more than what is contained in a total order. There is plenty of room for impossibility theorems here; that requires detailed research. But the point is that it is helpful to think in terms of different states of the voters' actual preferences, rather than in terms of different ways of sampling some infinite but unknown sets of preferences.