I think it's laymen rather than philosophers that are responsible for things being left as "cool mysteries." To philosophers, I gather, they're "open questions," things that either haven't been resolved yet because discourse is ongoing, or because philosophers have failed to find an answer. Or if they have been resolved, then they're questions presented by philosophy teachers to sharpen their students' analytical skills. But the questions rather than the skills trickle down to non-philosophers.

As a non-philosopher, I do personally like a cool mystery for its own sake, but my point, I suppose, is that since the question of "which one is the fork" reduces almost exactly to the Philosophy 101 presentation of the problem of identity over time, the same framework can be used for analyzing it. If I were to write a paper on it, my thesis would be that the question of "which one is the fork" is the result of a bug in the layer of semantics, and so what actually happens is determined by the lower layers of abstraction: code, law, and psychology.