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You're right that true and false are not binary (pun intended), but falsification absolutely does "work" in the sense that data which contradicts a theory should generally lead to a much stronger updating of priors than data which agrees with a theory.
Depends on the theory and what you mean by 'contradict'.
For theories about simple systems with a small number of axioms and moving parts, where the predictions of a model are very binary and precise, yes, it's easy to heavily contradict a theory. Put two spheres of a certain mass a certain distance form each other in an endless void, and your theory of gravity will very precisely state how quickly they should move towards each other; observe anything other than that, and with a complete absence of plausible alternative explanations for the observation, you've got a very strong update against your theory.
But in much more complex systems with millions of moving parts and interacting effects, such as a person or a society or an economy or etc., models tend to make much broader and more stochastic predictions, and there tend to be lots more plausible explanations for divergences from those predictions. And thus the updates from both affirming and 'contradictory' evidence tend to be much weaker, and more symmetric.
Of course, that's why a lot of people tend to get frustrated about studies of complex systems, and call that all 'soft science' and sneer at it. but this always seems like a type of cowardice to me; yes, studying those systems is a lot more difficult, but that doesn't mean that truths don't exist or can't ever be found, and they are often very very very important!
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Hmm, I was going to dispute this, but on reflection I think you're right, even if my exhausted brain isn't processing things very well.
The reason this works is because it's much easier to come up with a false theory that agrees with existing data than it is to come up with a true theory that appears to contradict existing data, so contradiction provides much more useful information for updating priors that confirmation does. And if your theory is consistent with all conceivable data (i.e. is unfalsifiable), then the theory doesn't have any predictive value because it is consistent with all possible outcomes.
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