I don't think that you will find my answer satisfactory, but does there have to be an alternative? I think that many syntactic mysteries are mysterious only because syntacticians assume that they work in particular ways. Once you start looking at these problems from the perspective of pragmatics, competing interpretations and conventionalisation, many of these mysteries disappear.

The point to make is that interpretation makes use of multiple sources of information, of which syntax is just one. You don't need an elegant, tight fitting, constrained syntactic structure to use language. Which incidentally is why we can understand things even they are not grammatical. I firmly believe that the issue of "grammaticality" simply boils down to convention. Not to mention that many constructed examples used to motivate formal grammar research (be it syntax or semantics) are borderline acceptable themselves to many native speakers.