Then, why don't smart people do the same thing? Instead of (or in addition to) competing with each other, why don't we insist more that being smart is cool (and being stupid is incool)?

To an extent, I think that does happen.

I think that part of the problem is that a lot of people think that "smart=successful=does well in school=the establishment figure", and that therefore people who don't do well in school tend to lash out against it for fairly obvious reasons. If a person who can't do well in school views himself as "not smart", then for reasons of ego-self-preservation he will tend to decide that "being smart isn't important" and lash out against things he associates with that trait.