The majority of private schools mostly have students who are at least average. There are a few private schools which are designed and specialise in students with a range of SEN's but they aren't the ones getting top results
State schools will have the entire cohort including those who are working very significantly below average and who are not capable of achieving C grades and above no matter how good the teaching is
Overall the range of SEN is much wider at state schools. There ARE many children with significant needs who are floundering in state schools with poor support, children who would probably do a bit better is special schools with small classes and teachers knowledgable in handling their issues. EBD provision is very poor in some areas - I'm LUCKY that my DD1 and DD2 were both able to attend an EBD specialist school, which is quite close to where we live.
I had to put up a fight to get my DD2 a statement and therefore her school. And she has/had (IMHO) very significant emotional and behavioural issues, although not extreme violence
I know many parents whose children are in mainstream state schools, and those children have needs ranging from autism to schizophrenia to Down's syndrome.
And I do think it's realistic to say that behaviour and emotional issues among students will always create a divide between selective private and state schools.