I'm sorry to disappoint you with an anticlimactic answer: I don't know.  But here are some of my thoughts:

The first thought I had was that they break down under intense pressure.  It's no secret that prodigies are under intense pressure from family, media, and fans, and, as famously illustrated in The Queen's Gambit, some turn to self-destructive habits to escape expectations.  While the rise of prodigies often receives substantial publicity, stories of their falls are also covered (to a much lesser extent), so it's possible that most prodigies drop off their pedestal before they reach their potential.

This is a plausible explanation, but it doesn't account for prodigies that continue their success into adulthood.  The success of highly televised prodigies like Kim Ung-Yong and Jennifer Pike has skyrocketed through the years, but still, we never hear about them now...

...or do we?  You probably knew that Pablo Picasso was drawing before he could talk, but did you know that Jean Piaget was a world-renowned mollusk expert as a teenager, or that Marie Curie specialized in language and mathematics at the age of four?  Both Curie and Piaget went through a brief boom of fame in childhood before entering the global spotlight later on in life ― however, few know their earlier life achievements, despite their accomplishments being arguably as equally impressive as other unknown prodigies.

Here's my take: the child prodigies we hear about, the ones that make national news early on in life, reach a point where they can't keep up with expectations, even if they want to ― a five-year-old composing his own music is sensational, but a twenty-year-old doing the same is simply average.  That's why the adult achievements of well-publicized child prodigies are ignored, but similar achievements of less-televised child prodigies make world headlines.

So, the next time you see toddlers playing chess on national news, cut them some slack ― they're human too, and they're likely not going to live up to the hype.