Is it worth putting small kids in competitive chess? I saw some chess tournaments with kids under the age of 7, and even under 5!
Surprisingly, including their performance.

Is adding the pressure of competition at this early age worth it?
If the interest, competitive drive, and knowledge of the rules are there, then I wouldn't recommend against it.  Even if they play competitively and find their talent for the game perhaps to be lacking, they can still take away much from the game, I feel.    

Personally, I was pushed hard when I was young at the game (started at age 5, pushed relatively hard until about age 10), took a lackadaisical approach to the game as a rebellious result, but kept playing competitively through high school.  I didn't do so bad, but not great either.  What I took from the game though provided a niche for me for a time.  Even if not used in any professional sense it can provide a potential life-long hobby.  Even if it provides neither, denying them the opportunity given a desire to play competitively seems bad.  It's just how one prepares.

Try and get an idea early of how good your child is.  Playing with them a lot outside of tournaments first is a good way to do this.  Then, you'll be able to gauge how hard to push them.  If they're really good, push them just a little bit.  If not, don't press them to play at all.  Either way, if the desire remains to play and they're doing well in school, why not?  I know I always had the desire to play competitively, and I'm glad in ways that I was pressed to play competitively.