The interviewer is implicitly trying to figure out if you could help him/her look good in front of the client and get promoted/elected to the next level. How do you do that? By showing you can follow a structured, repeatable and MECE approach to solving a problem, and draw the logical insights.
If you crack the case because you came with a battery of questions seemingly at random and lucked out, you do not advance. If however you follow a MECE framework and run out of time, you still stand a chance of moving on to the next round.
Ben is exactly right: unstructured case means going home; structured case is a must, no 2 ways around it.
Expert case by