It might be of some general interest to note that there exist such objects - distributions over distributions, or metaprobabilities. They have practical use - for example, if you are playing a game with uncertainty, you might observe something in the next step that changes your belief, or distribution over world state, b = P(s). If you have some expected distribution of next observations then you can talk about the probability of having some belief in the next state. Roughly P(b'|b) = P(P(s')|P(s)). You can collapse this into simple probabilities if you just care about your new observations, P(obs'|b) = \sum_s P(obs'|s)P(s|b).
I could imagine some frameworks where confidence intervals in this way would be useful - ex. I have 3 fairly different world models that I think are equally likely, each has a P(election), what's the P(election|world) and get confidence intervals across world models rather than just summing over them to P(election). But I agree that for most common approaches that simple probabilities are most useful and clear.