Nothing, based on the existence of thousands of exactly such schools within the US alone.

On the other hand: a disagreement about the actual definition of gifted, based on the existence of thousands of such schools in the US alone. "Gifted" in some jurisdictions simply means something anodyne like "top 10%" which obviously doesn't get close to creating an actually targeted school environment for your Mozarts.

It's controversial in education schools to "track" students, i.e. sort them into ability-categories and tailor each category's experience to its needs. For example, activist groups in New York City have been trying to kill gifted-and-talented schools and programs (e.g. Bronx Science high school) for years. It's painful to watch.

People can and do create rigorous private schools, but they're not accessible to the masses and often embody the same anti-talent mentality public ones do.

Communists.