FWIW, I tend to think about it the same way as you.
My sense is that the difference isn’t important as long as you’re willing to force all individuals to use the same subjective probabilities as are used in S.* As Weymark notes, Harsanyi’s axioms can’t in general be satisfied unless this constraint is imposed; which suggests that the approaches are the same in any case where the latter approach works at all. (And Aumann’s result also tends to suggest that imposing the constraint would be reasonable.)
* If U_i is the expectation of u_i(x), and S is a (weighted) sum of the U_i, then S is also the expectation of s(x), where s(x) is defined as a (weighted) sum of the u_i(x), provided all the expectations are taken with respect to the same probability distribution.
FWIW, I tend to think about it the same way as you.
My sense is that the difference isn’t important as long as you’re willing to force all individuals to use the same subjective probabilities as are used in S.* As Weymark notes, Harsanyi’s axioms can’t in general be satisfied unless this constraint is imposed; which suggests that the approaches are the same in any case where the latter approach works at all. (And Aumann’s result also tends to suggest that imposing the constraint would be reasonable.)
* If U_i is the expectation of u_i(x), and S is a (weighted) sum of the U_i, then S is also the expectation of s(x), where s(x) is defined as a (weighted) sum of the u_i(x), provided all the expectations are taken with respect to the same probability distribution.