How does private schooling basically work everywhere?

In the US (and pretty much everywhere else in the world, I am European), you have your public schools that are managed, financed, and staffed by the government. The quality of public education varies greatly from town to town, city to city, and state to state. <br><br>Alongside public schools, you also have private schools: institutions run by a private organization, company, or individual, which get to choose who they accept, what they teach, how they teach it, who they hire, etc. Generally speaking, private schools are more expensive than public schools and are usually considered to offer a better education and better environment for the kids. <br><br>Ok, fair enough. <br><br>Now, wouldn’t it make more sense (financially, organizationally, and socially) for EVERY school to be state run? For the government to ensure that their citizens have access to education, especially at a young age, and for those institutions to be as uniform and equal in quality as possible? It seems like it’s a pretty crucial thing for society to invest into, like police or military.