I don't know if a lot of thought goes into philosophy of the weeding-out questions. The profs know that they have too many people in the program, so they want to cut the numbers down to those able, by hook or by crook, to do the problems. Whether they would actually make good physicists is a question to be answered later (when they need to choose advisers and be graded on their qualifier and dissertation). Engineering schools seem to go through the same process.
On the hockey stick problem - that's a pretty good one. But not one I'd want to see on an exam! I guess I'd attack it by starting thinking about what happens when the puck hits the very end of the stick (pure rotation about its center of mass), and what happens when it hits the center (pure translation of the center of mass). But if the puck transfers all of its energy in the first case, then it will stop while the stick rotates, so presumably the puck will be hit when the stick rotates around. At that point, will all of the stick's energy transfer to the puck? I'd think not in practice, but on a frictionless surface, maybe so. That case is probably a variation of the balls-hanging-from-threads see-saw "Newton's cradle" toy.
It's more complicated if the puck hits between those two extremes - being a combination of translation and rotation.
It's the type of problem one could get lost in on an exam... :-(
I'm glad I don't have to think about those things any more!! :-)