One of the fundamental factors that makes private, fee paying schools more successful than state education is that state schools are limited by the government in how they can spend their money, and in that they are required to stick vigorously to the national curriculum. Allowing state schools to become, by the original definition of the term, independent, would allow them to spend money in the way that any educational establishment feels is best, rather than the way the government feels is best. Currently, the state system treats all schools identically. It's like testing which animal is the best by getting them all to climb a tree. The 'free schools' that the Conservative government is currently creating is exactly what this country's educational system needs. Independent schools, private schools that are in the state sector and funded by the government. State run schools that have the independence to run and spend as they see fit. Of course I'm not saying that we should privatize schools in such a manner that every student is required to pay fees, but privatization in the way of freedom of the strict, government control that state school are currently subject to.
Privatizing schools is the best and cheapest way to make sure our children get the best possible education they can get. Privatizing the schools forces them to compete with each other which encourages them to do better. It would also allow variety in our schools. There is no reason we can have religious school, schools which require uniforms, school that practice corporal punishment. Then a parent can truly choose what school they want their child to attend.
Private companies are designed to make a profit. They build their entire infrastructure to generate money. The public education system can generate their entire infrastructure in the same exact way as a private company. The great thing about public education is that they don't need a profit, so all the extra money can be used on something else.