Take violinists, for example. Essentially every single world renowned soloist was "some sort" of child prodigy. Now, I've heard some soloists argue that they were not, in fact, child prodigies. For example, may favorite violinist, Hilary Hahn, has said this. She still debuted with the Baltimore Symphony Orchestra when she was 12, and here she is performing as a soloist at 15: https://youtu.be/upkP46nvqVI. Nathan Milstein, one of the greatest violinists of all time, said he was "not very good until his teens" - he still started playing at the age of 5, and at the age of 11 Leopold Auer, a great violin teacher, invited him to become one of his students, so he clearly saw his potential.
I have no doubt lots of prodigies burn out. But, at least in the world of violins, essentially every great soloist was playing at an extremely high level by the time they were in middle school.
I had a friend who could play all the Chopin Etudes at age 9. Some of the best art simply requires a virtuoso to bring it to life.
Justin Bieber clearly was that. His youtube videos got him discovered at age 13-14.
Vanessa Paradis made her first public appearance as a singer at age 7.
There are several children prodigies I've seen on YouTube (singers, drummers, guitarists). They clearly have such talent that even at young age they do music better than most people would do with infinite amount of practice.
As to your question, the prodigy is, by definition, extremely rare. They clearly exist (Bieber, Paradis) but, by definition, you can't expect to have a lot of them.
And "why aren't 7 year olds headlining for Taylor Swift" is not a fair bar.
There are reasons 7 year olds don't do world wide tours that have to do with things other than musical talent. Like being in school or not being allowed to take a bus by themselves.