It is understandable that this will take considerable effort, and thinking, to sink in.Hmmm ... I guess I see what you are saying here ... since a single particle must necessarily give single-valued answers, the measurement uncertainties are not relevant to the HUP. Instead, it is the predictability of the result that comes in to question. In classical physics, if we knew the momentum and position, we could predict the future trajectory with arbitrary precision. However, in Q.M., since the particles can only be measured once (meaningfully anyway), the only way to test predictability is with multiple experiments. Even so, is it correct to say that the HUP doesn't come into play for the single measurement, because doesn't it define the range of possible values for the measurement?
The HUP does come into play for single measurement, but not with regards to the uncertainty of that single measurement. Let's say that you want to measure the momentum, given that the particle has passed through a slit of a certain width. You measure, say, p1, with an uncertainty dictated by your instrument as Delta(p). Now, your next value will be p2, and how well you can predict p2 depends very much on the HUP, i.e. the width of the slit. But p2 still has the same uncertainty in that single measurement, i.e. Delta(p) (assuming your detector is uniform).
So yes, the HUP comes into play in how spread out each of the values of the momentum (the p's) that you measure, but not in the uncertainty of each of those single measurements.
Hopefully, that makes things clearer.