>>The goal of this is too test whether the interviewee can recognize that this is a case to apply this algorithm in.
That's the easy part. Everyone knows if you have sort a large data set, then you partition it in someway. Searching a huge data set requires building a tree on some criteria before hand and eliminating whole branches at every node. Or that they have to use a graph algorithm to figure out shortest paths and cost associated with the paths. Or that if you face a situation where you recompute many results you memoize. Or that writing nested for loops is a very bad idea. These are easy things to spot.
But they won't clear you in the interview if you don't get the mechanics right. Knowing BST helps in sorting doesn't cut it. They want you tell them how you would traverse across a tree in some 'level ordered' fashion. Or list different ways you could order a graph and list the exact heuristic. From here on this is pure memorization.
There is huge difference between inventing a new framework of thought even at an abstract level, and knowing the precise mechanics of balancing a tree.