I am probably repeating what has been said, but I don't think this is a question which can be answered out of context.
It depends on where you live and the schools that are available.
I'm a teacher who has taught in the independent and state sectors. Both my DCs went to all state schools because we couldn't afford otherwise.
Their primary schools were very good, and DDs secondary school was almost on a par with some local independent schools. Both of my DCs had private tutors for A level and ended up at Russell Group unis, gaining good science degrees.
Had we paid squillions for private schools I don't know if we'd have had abetter outcome.
DSs school mis-informed him over entry requirements for top unis re. A level subjects and he ended up not being able to apply to the top 4 which he wanted, and one teacher for 1 subject was useless meaning DS ended up with a B not an A grade. So a few regrets there, but some of his peers got into LSE/ Cambridge etc.
But the cost of private secondary schools is high- you are looking at £20k each which is £40L pa after tax- equivalent to almost having to earn / have £75K gross to pay for fees alone- each year.
As a teacher I have tutored children for common entrance ON TOP of what their prep schools offered, and it's not unlikely that a child at a state school could be tutored for CE by a tutor.
So I don't believe the situation is so clear cut as some posters here maintain.