Private schools flourish because they are the preserve of the wealthy. There is nothing more to it than that.
Bit more complex than that
Firstly you have to really want your child to have a "good education" to sedn them there
As a result you are more likely to make sure they do their homework and get good marks - and they will probably go to the parent's evening and check what the teachers say
On which point - if the kid mis-behaves then the school can be kick them out quite easily and without going through the multivarious steps that a normal school has to go through
Also - as there are fees set by the school then they can have smaller classes - ask any teacher and they will tell you that smaller classes equals better teaching and better results
and happier teachers and kids
chuck all that together and you get better results
BTW - I went to a fee paying school - the facilities were rubbish - floorboards so odl that the nails stuck up an inch or more as the wood had worn down around it
Sports facilities were a couple of fields for rugby and cricket and a "gym" that was mostly one big room with space to volleyball, basketball and stuff - all using ancient equipment that had been patched up over many years
But parent engagement was excellent and that made a lot of difference
N.B. I have also taught in an ex independent school that was very similar - very good parent engagement and attitude of the girls (single sex school) - many of whom had had the female member of their family attending the school for several generations
of course - the wealthy are far better able to be able to afford that