I think my skills as an experienced developer come from having solved many problems (as well as having made plenty of mistakes) in the past. When I look at a problem these days I try to think back to similar problems in the past, what worked and what didn't. Its not so often that actually need to sit down and think through a new algorithm. So it comes down to "remembering things" as much as it does to "working things out".

There must be an in-between case where a small problem is solved by remembering the method. The kind of thing where you end up looking up the double-angle formulae or checking the definition for integration by parts. I guess the programming equivalent is remembering how to insert into a tree or do a bubble sort.

Problem-solving in the general case encompasses all of human creativity: I wouldn't claim that one becomes capable of tackling difficult problems through memorising techniques. But once you're part-way to solving a problem there is often a sequence of simpler steps (analysis, not synthesis) - surely most of us would like to be faster/more efficient/make fewer errors in these steps.

It can also be used to schedule review of problems.

You can enter homework problems (like page number and problem number) into Anki and use that to schedule review.

You can use it to review proofs, too.