What is the role of kōans in contemporary Buddhism? I have a copy of The Iron Flute that I use to study individual kōans, and read the analysis provided.  I feel like this provides me with some good insight, but it also feels... backwards.

How are kōans used in modern Buddhism?

Are they intended as teaching aids, where the explanation illustrates one or more principals, or are they intended as puzzles that only sufficiently advanced students should be able to figure out?  Or can either strategy (or altogether different strategies) be employed to gainful effect?
The way my Zen Master used them (as much as I can pretend to have understood my Zen Master at all!) is to essentially give the person a hint about a particular pathological pattern the person was stuck on. So the koan would always match the elephant in the room that the student was working hard to ignore. The student could then resort to analysis, or simply mull over the koan while going along with the daily activities. In any case, solving the koan required getting an insight into oneself and one's fundamental relationship to the world, as opposed to figuring out some new external fact or gaining an analytical understanding.