I agree that the US system teaches mathematics in too formulaic and granular a way. The way I see it, a good way to divide math students into two categories is to look at those who will use math in their future careers as scientists, engineers, etc, and those who will not and are only studying math to "see the world from another angle".
In either case, a spitting-back-of-formulas approach is counterproductive: Scientists and engineers need to (or at least would do very well to) understand how the mathematics they use works. On the other hand, those who will not use math in the future have no real need to pack their brains full of formulaic tidbits; they will get the most value through pursuing a problem-solving approach that generalizes well to other areas of life.
In short, no matter what the class's level, mathematics definitely needs to be taught from a more problem-solving, integrated perspective. "Pre-med math" that consists mainly of memorization and regurgitation does nobody any good.
The main problem, I imagine, would be recruiting teachers that can actually teach mathematics in this way. That raises a whole different question, though - why do we have "teachers" and not mathematicians teaching math; why do we have "teachers" and not historians teaching history? It seems that much of what education school teaches is how to keep order and run a classroom of bored and antsy students, but wouldn't it be more effective to simply teach interesting material and keep the kids engaged?