It is curious that we definitely have degrees of belief, but we usually can't precisely introspect them. I would guess this is because introspecting the reasons pro/con a belief, they're temporarily made more salient. So we have a feeling of vacillation.

I don't find ranges of probabilities of final binary outcomes (for making decisions) to be useful at all. A single number is all I need. Just because this number may change as I focus on different pieces of evidence, doesn't mean that it needs to be represented as a range. But if your model contains inside it parameters that represent the probability of some latent variable, then you should indeed be integrating over the distributions of all those parameters.