Is this really the case? It's doesn't seem true of axiomatic approaches to decision-theory in general, so is there a special reason to think it should be true here?

I just gave the reason. "Some mechanisms will result in always choosing actions from one category; some mechanisms will result in sampling from different categories proportionally to their votes."

The aggregation mechanism is a lot like the thread priority system in a computer operating system. Some operating systems will always give the CPU to the highest-priority task. Some try to give tasks CPU time proportional to their priority. Likewise, some aggregation mechanisms will choose the most popular option; some choose options with probability proportional to their popularity, never giving any voice to minority opinions. You have to choose which type of aggregation mechanism to use. But this choice is exactly the sort of choice that the parliament is supposed to be producing as output, not requiring as input.