The experience of the professor at Harvard in the video contradicts your last paragraph. They know how to apply the math but they don't understand the problem. And the traveling problem I gave underscores this point. It's a very simple problem and almost no one gets it correct.
If all students are taught to do is "memorization of problem types" (to borrow GP's term), then they will only be able to handle actual problems that match one of the necessarily finite set of types of problems they have memorized. A different selection of problem types to memorize will not provide general coverage -- that requires learning to construct a solution to a new type of problem.
Nowhere in the thread has it been demonstrated that it would be impossible to design a word problem that does require the construction of a solution.
One of the problems I had with my dif. eq. class was that the grad student teaching the class just focused on "tank problems" and other "x problems" sets. So I see why story problems alone are not helpful. But, if they're judiciously used to encourage the understanding of abstract mathematical concepts, I think they are invaluable.
