Yes, we know that preference involves more than ordering and that "strengths" are relevant to decision making. We know this because it explains an obvious empirical matter, that there are scenarios like the one you cite where we prefer X over Y, but end up choosing Y over X. Not a puzzle; not a paradox; how it is. And of course the way we tease out the "strength" of your preference involves, e.g., looking at how your choice varies between counterfactual scenarios involving different probabilities.
But the question is whether a "score" given by a voter on a ballot indicates anything psychologically interesting, and, more importantly, whether it indicates anything relevant to the "judicious" selection of a candidate. It seems to me that, empirically, the only meaning we can _seriously_ assign to a "score" on a ballot is a relative-scoring-as-indicative-of-relative-preference. I don't know how "10 to A, 20 to B" could indicate anything beyond the simple fact that the voter prefers B to A. And surely my "10 to A, 20 to B" needn't have the same meaning as your "10 to A, 20 to B" -- I think all we can really say is that we both prefer B to A. What else?
Now, you might want to try to interpret these "scores" as something like a hypothetical question put to the voter about how much they might pay to have this-or-that person elected; but if you know anything about self-reports of this kind about hypothetical scenarios, you know that people are horribly inaccurate (either intentionally or not), and that the setup is contrived. Serious measures of preferences involve actual stakes, markets, etc. Good luck with that here.