If we assume that all theoretical choices are possible with positive probability, then the set should be, in fact, smooth, or at least differentiable, as it would be impossible for two different weightings to produce the same exact outcome. This makes the result nicer.

I proved an essentially analogous result, viewing it as a justification of utilitarianism.

I think the fact that Pareto is equivalent to utility-weighting is well-known in economics. It's one of the consequences of the separating hyperplane theorem. I have not seen it applied to bargaining, because economists mostly think in terms of actual rather than optimal bargaining. Perhaps it has been, though.