I know several students who have USAMO, and USAPHO were not even admitted by Cal.
So if you think your children are really interested in Math and Science, by all means push forward. Otherwise, let them freely explore may not be a bad idea.
MIT, Caltech, Cal, etc will take only a few from the each competition category. But they will take a lot of more students from other pools. Standing in the shoes of AOs of MIT, you would know what to do.
I encourgage my children to do more exploration of their interests. Would you like to be the founder, or CEO of Google and Microsoft, or would you like to be the top coder in the organization? We, as a race, should not waste our children's time on competitions. Think of large pictures. If they are not admitted by the top colleges, we need to work on the policy so that the colleges take more qualified students instead of letting our children to beat each other out. That means that, we, as parents, are " weak". We don't have enough influence and power. We need to work harder make changes, not deprive our children of their childhood, and their sleep.