I agree with the claim that problem solving is essential to learning mathematics. And I have had the exact same experience where I did well on math exams by ignoring most of the theory and proofs of main results, and focusing solely on examples and problem solving.

However, I think straight-up memorizing definitions and theorem statements is really useful for problem-solving/exam prep, just so they're at your fingertips. There's no way you're passing a real analysis exam if you can't regurgitate the epsilon-delta definition of a limit in your sleep.

What seems to occur (at least for me) is that you naturally memorize all of these things in a somewhat inefficient fashion by doing problems. If the concept gets used in enough problems, it slowly burrows its way into your memory - and this is a very durable kind of memory, as you point out. But I do think for things like graduate school qualifying exams you can "juice" the process by explicitly memorizing core material.

Probably it's not as useful for doing research, though.