Should high school math contests be banned? Absolutely not. I agree with what Ted Alper said about the negative aspects of the "pecking order". At that age, even if students are mature enough to understand the distinction between "talent" (whatever that means) and "hard work," so many factors other than "hard work" determine how well you will perform in competitions. As a result, many students reach a point where their performance plateaus, yet they feel they are working very hard. So they attribute their plateau-ing performance to a lack of mathematical ability, when in fact, the problem may not be mathematical in nature at all and thus much harder to find (test anxiety, mental inflexibility, poor organization skills, poor problem-solving habits, etc.) And of course you have all the students who buy into the talent theory.
The other thing that I found to be a bit of an issue with math competitions was the overemphasis on how they were about "creativity" rather than "memorization". This paradigm exaggerates the difference between contest math and classroom math. While classroom math does mainly boil down to memorization, contest math is hardly anything close to full-blown creativity. Even at the level of the USAMO/IMO, the vast majority of problems are still best solved by pattern recognition, a relatively large bag of tricks, and sequential deductions. A contestant who believes too much in the efficacy of creativity will spend too much time trying to come up with "the clever thing you need to pull out of your ass" to kill a problem in one blow (my friends and I refer to this tendency as "eyeballing the solution"). I believe this is the main reason why many USAMO contestants get to the point of solving the "one-step" problems 1 and 4 rather quickly, but often never reach the point of solving the multi-step problems 2 and 5. At least for me, I did much better on the USAMO with relatively little extra practice once I abandoned the notion that contest math was largely about "creativity" and "eyeballing the solution."
From what I understand (based on personal experience and anecdotes of friends), Red and Blue groups (the two lower groups at MOP) had a big emphasis on "creativity," while Black MOP (the top group) had a nontrivial portion of the lectures devoted to dumb tricks. For example, in 2010 (I think?) Black MOP had one conference dedicated solely to "dumb-assing" (multiplying out everything in an inequality problem and using weighted AM-GM to clean up the pieces) and one to coordinate bashing. Both are the complete opposite of "creativity," but apparently they were included because the instructors thought them beneficial to IMO performance. I believe that it would be prudent for students in the competition track to be aware of this early on so that they do not overemphasize the role of "creativity" in contest math (and for that matter, real life).
That being said, I would not change the contests one bit - just wanted to point out what I perceived to be a few potential pitfalls (and hopefully help out a few of the younger members here who are still participating in these). But perhaps I was just stupid back then, and the rest of you guys already know this stuff going in.
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