> Wrestling with a problem for long periods of time is not just a convenience for the teacher, it is a skill that will serve students well for decades to come.
And one of the best ways of developing that skill is... learning higher-level math. This can also include 'competition math' topics of course, but they should be approached as self-contained subjects of their own, not just as a bundle of disconnected "tricks" to be applied solely in a competition- or puzzle-solving context.
Depending on how the course is set up, maybe. Most math courses are not set up to make students wrestle with problems for extended periods of time, even through University level.
I took courses in topology and number theory in undergrad that were set up this way—the professor did almost no lecturing; we were given a series of results to prove and expected to wrestle with them ourselves (mostly alone as homework). Once you thought you had a proof you presented it to the class. But this is very atypical. Your typical calculus or differential equations or linear algebra course does not develop this skill.
And one of the best ways of developing that skill is... learning higher-level math. This can also include 'competition math' topics of course, but they should be approached as self-contained subjects of their own, not just as a bundle of disconnected "tricks" to be applied solely in a competition- or puzzle-solving context.