Do they not teach the trick to expand 5x to xxxxx? I still use that whenever I'm in doubt.
A few years ago I tutored high school kids in math and physics and I wholly agree with your points. I had a few strategies that worked well depending on the kind of student. One was to break a problem into its smallest constituents, to ask basic questions about those problems (which they usually got right) and to then assemble it into a bigger picture.
Another is to attack a problem like you're Sherlock: what do we have here, and what do you know about problems that look like this? In my opinion, getting a student from doing math to understanding math is by asking the why question again and again and patiently teaching them the underlying fundamental principles.