Recently I was impressed to see that the Winograd schema challenge -- a particular challenge dataset of the kind you mention -- saw huge performance gains from statistical NLP methods, which implies that the statistical methods encoded (something that at least sometimes works a lot like a substitute for) common sense about the world and the context of utterances.
The Winograd schemas were my favorite example of "XYZ" here. They involve pairs of sentences where the sentence is parsed differently based on assumptions from external-world knowledge. For example
Putting the barbell on the glass table isn't a good idea, because it's too (heavy/flimsy).
What is too heavy?What is too flimsy?
Or, from memory rather than finding the original version
The local authorities refused a permit to the demonstrators because they (feared/advocated) violence.
Who feared violence?Who advocated violence?
The compiler confirmed that that the program was buggy because it (enforced/failed) typechecking.
What enforced typechecking?What failed typechecking?
The doctor recommended surgery for the patient because she was (suffering from/experienced with) coronary artery blockage.
Who was suffering from coronary artery blockage?Who was experienced with coronary artery blockage?
In these cases, the most plausible referent of a pronoun changes depending on the semantics of the other word that you fill in, for a reason that has to do with the outside world, not syntax. (I don't each case absolutely has to include a "because" clause, but it's the simplest way to supply disambiguating context.)
I totally thought statistical techniques would be horrible at this. There's good reason to think that these require some detailed knowledge about the world. The statistical techniques were horrible at this, for a while, and now they're abruptly great at it (if I remember correctly, at or above typical human performance).