They reject the problem kids who end up at schools with a "take one and all" policy. My school was the same, it was an ex-Grammar School trying to preserve its reputation by selecting the best pupils from its catchment area.
Some do, some don't. I'd accept that as a critique of your vision of egalitarian schooling. There are arguments for and against. You could apply similar rationale for single sex schools.
Not having a go here but when I lived in London I never noticed anyone with any morals which I had instilled in me whether it was beaten into me or not but people I came across didn't seem to know the difference betwween right or wrong so that attitude comes from a early age and education has a big part to play for the rest of your life.
I tend to think that faith schools run the risk of exerting a virtue on its pupils that leads them to believe themselves to be genetically better than their non-faith'd counterparts.
"I believed in father christmas I looked to the sky with excited eyes `til I woke with a yawn at the first light of dawn and I saw him and through his disguise" Wise words people, wise words.
I believe in a thing called love Just listen to the rhythm of my heart There's a chance we could make it now We'll be rocking 'til the sun goes down I believe in a thing called love Ooh! Guitar! Wiser words people, wiser words.
I expect each of your institutions takes in a handful of special needs children in order to hold them up as an example of what sin can create or some **** like that.