I feel you're missing the point the parent was making. The point as I see it is not to try not to know the answer without deriving it but to not set out to simply memorise the answer.
Of course through use you'll commit certain proofs, formulae, corollaries or what have you to memory; but that's not what people generally mean by "memorising". When studying [undergrad honours, UK] I didn't set out to memorise things and often relied on working out formulae/results/theorems from first principles or from some other theorem. [Perhaps this is a physicist thing, I did joint honours in Phys/Maths]
Remembering and memorising are different approaches that lead to a similar result; however IME remembering when you didn't set out to is often much better. For me I could cram and memorise some results, but then that subject matter never stuck around long afterwards.
For my own studies, there are many many definitions and theorems I just have to memorize. It is true that I have familiarity with how the theory is all proved, but in the end I have to remember what all the main theorems say. For instance, there is no way you can figure out what "Alexander duality" is or the axioms of a "group" are by their names alone.
Re computational errors: a lot of times they tend to be hidden conceptual errors, since if you had understood the concepts better you would have detected the error!
Re hard problems list: I disagree that keeping such a list amounts to memorizing certain kinds of solutions. I try to keep one mentally, which I use to try to figure out what about the problems seem hard. The hope is that the hardness of the problems dissolves, leaving me with fewer problems and more insight. Even better is when the hard problems suggest other hard problems.
Re "it is clear that": I recommend you find a grading job on campus. The usual reasons this "works" is that the grader is overworked and forgets it is only clear to them, or the student didn't say any nonsense elsewhere so they've earned the benefit of the doubt. But don't delude yourself outside the exam. (The worst are students who make their work intentionally messy to try to throw off the grader. Believe me, the mess is transparent.)
I used to have a distaste for memorization, believing in an ideal of "real math" which was some fountain of pure understanding we might come to know, but over the years I realized that to even begin approximating that ideal you need fluency in the various languages of mathematics --- and to learn a language you need to memorize its vocabulary at the least!