As others have said, with pupils being selected through entrance examinations, better facilities and smaller classes, it isn't surprising that private schools yield better results. It takes a gifted teacher to keep the attention of a largish class of pupils of varying abilities and backgrounds, and I think there are probably more of these in the state system than we are aware of.
Some private and state schools do share resources but there is criticism that many do nothing or the absolute minimum , such as allowing limited use of their sports fields or swimming pools.
Children don't necessarily have to be that clever to pass an entrance exam. They just need to be properly prepared to pass it, through attending "feeder" prep schools that know what will be required in the entrance examination and "teach to test", and/or through private tuition. I think it is different for children who are applying for scholarships - their marks have to be well above average.
Other countries that have excellent educational standards, such as Sweden and Finland, have only a few fee-paying schools. Germany, which reputedly has a very good education system, only had 4.8% of private schools. Since austerity, this has risen to 7.7% as cost-cutting measures have negatively affected state schools. It will be interesting to see whether these changes benefit the country as a whole or whether it reduces the educational attainment of the majority of young people.
I think people should care about what happens to all children - not just their own - and a system that reinforces class division and perpetuates a system where privilege often takes precedence over true potential is ultimately no good for anyone.