This suggests that the right formalization of "bargaining power" would give a nonnegative scaling factor to each member of the group, and the group will act to maximize the sum of the agents' expected utilities weighed by their respective scaling factors. ... Of course, at this point this is merely an intuition, and I do not have a worked-out proposal nor a careful justification.

There is a justification for that intuition. Some have objected to the axiom that the aggregation must also be VNM-rational, but Nisan has proved a similar theorem that does not rely on the VNM-rationality of the collective as an axiom.