The university system is mostly a mixture of choice by merit and choice by money. The private K-12 is similar. There are also merit-based programs within the public school, especially high school.
Neither you or Mr. McCain is proposing anything to help the social-economically disadvantaged students to perform better.
The competition does exist in public school. You just vote with where you live. Are you suggesting to use a merit based system to decide who goes to which public elementary school? I don’t think it is a bad idea but it will not happen.