Why does a bouncy ball impart more momentum in a collision than a sticky ball? And 2022 AP1 #4

Today was the first day of physics fight prep.  One of the questions to investigate was the paragraph problem (#4) from the 2022 AP Physics 1 exam: A ball of clay sticks to a block, and a rubber ball bounces off the block.  

The important bit of this question is to understand that the bouncing ball causes its block to move faster after collision than the sticking clay does.  How do you justify this result with physics facts?

The wrong way, that I saw many times both today and at the reading, says p=mv; since momentum is conserved, the p is the same after and before collision.  Since the clay sticks, that causes the block to have more mass after collision, and thus less speed.  

This wrong-way approach makes an incorrect comparison.  We want to compare the speeds of the two blocks in two different collisions.  It's not true that the block has the same momentum after collision either way!  It *is* true that the momentum of the block-ball system after collision is equal to the ball's momentum before collision.  All this wrong-way justification is telling us is that for the clay collision, the speed of the block-and-clay after the collision is slower than the speed of the clay before collision.  Not what was asked.

One correct approach is to consider the impulse - the change in momentum - of each ball during the collision.  Because the rubber ball bounces, it changes its momentum by more than the clay does.  Then, conservation of momentum says that total momentum of both objects can't change... meaning that however much the ball changes its momentum, the block changes its momentum by the same amount.  The rubber ball causes the block to also change its momentum by more than does the clay does, meaning the rubber ball's block moves faster after collision.

The other correct approach looks at the vector momentum after the collisions as the same in both cases - pretend this is 10 Ns to the right.  When the clay and block stick, their individual momentums add to the total momentum - pretend you get 9 Ns for the block and 1 Ns for the clay.  (Any numbers work here: watch.)  After the rubber ball bounces, the momentums of the block and ball are in opposite directions and so subtract to the total.  Whatever numbers you choose, the block must have more than 10 Ns of momentum, because you must subtract something to get 10 Ns total.  (11 Ns - 1 Ns works, but so does 12 Ns - 2 Ns, etc.)