I'm not sure you need uncertainty to discount at all - in finance exponential discounting comes from interest rates which are predicated on an assumption of somewhat stable economic growth rather than deriving from uncertainty.

As you point out, hyperbolic discounting can come from combining exponential discounting with an uncertain hazard rate. It seems many of the studies on hyperbolic discounting assume they are measuring a utility function directly when they may in fact be measuring the combination of a utility function with reasonable assumptions about the uncertainty of claiming a reward. It's not clear to me that they have actually shown that humans have time inconsistent preferences rather than just establishing that people don't separate the utility they attach to a reward from their expectation of actually receiving it in their responses to these kinds of studies.