It looks to me like we've bridged the gap between the approaches. We are doing the same thing, but the physics case is much more specific: We have a generating function in mind and just want to know its parameters, and we look only at the linear average, we don't vary the powers (*). So we don't use the tools you mentioned in the comment that started this thread, because they're adapted to the much more general case.

(*) Edit to add: Actually, on further thought, that's not entirely true. There are cases where we take moments of distributions and whatnot; a friend of mine who was a PhD student at the same time as me worked on such an analysis. It's just sufficiently rare (or maybe just rare in my experience!) that it didn't come to my mind right away.