This may be true of Eton and Harrow but it is certainly not the case in the many small independent schools across the country. I have taught in both the state and independent sector and in the latter, salaries were lower and facilities Dickensian compared with the local comprehensives. And yet the small independents achieved excellent results.
The key factor is not the facilities, nor the teachers. Nor is it the pupils - in small independents entry is usually determined by ability to pay and, as someone once remarked, "There are plenty of dumbos in Volvo- land."
The key factor is the parents. They look for a disciplined environment where their children will be safe and happy; they monitor their children's progress day-to-day; they demand that homework be set and marked; they complain about poor teachers and object to their child sharing a class with disruptive pupils. If they are not satisfied, they can simply walk away, taking their fees with them - which keeps the headmaster on his toes.
Education is a cumulative, water-on-a-stone business. In the independent sector, no time is wasted and, over 11 years of compulsory schooling, this is what makes the difference.