Why do faith schools tend to do better than non-faith schools?
Why do faith communities tend to be more ordered than non-faith ones?
Well the clue is in the name 'faith'. 'Faith' is used to all and every religion, and to make a distinction between religion and non-religion based things - and this puts the answer in plain sight.
The important thing about 'faith schools' and 'faith communities' is not 'faith' at all - in my experience, most people who classify themselves as having a religion are very dubious about the existance/form of what their 'god' is supposed to be. You hear "well I'm christian, but...", "my family is jewish", "I was raised a catholic".
The common thing about all these 'faiths' is a common set of rules - the religions themselves. A clear set of rules, set out, explained, examples given at great length, and a history that they have tended to favour 'well ordered' societies (those that don't, have - by natural selection - fallen by the wayside).
What makes these 'religions' as distinct from purely moral/social codes is that that final arbiter is cited as 'god' an external judge who has no reason to be anything other than perfectly fair.
I suggest the 'faith' bit is most relevant to children and newcomers - like training wheels on a bicycle or scaffolding around a new building - it supports the subject while development is taking place - but there comes the time when the scaffolding is irrelevant, the development is complete and will stand or fall on its own merits - the scaffolding can be removed or left it place, it makes no real difference.
In the end, broadly following the rules of any religion is as good (often better) than trying to follow any other set of moral/social rules - even if you don't have faith or believe in its god.
So a 'faith school' does better than a non-faith school even when very few people there actually have any 'faith' - it is simply that they have a clear set of rules that are well known, easily remembered and that participants have agreed to follow - rules which cannot be challenged on a whim nor need endless explanation - the answer to challenges is "it works" or "god says so", if you don't like it then 'goodbye'.
Clearly a common moral code works well for societies, simple tools like doing the 'right thing' because you are expected to behave as if your every thought/action is being observed (by a god) are useful. Its 'religious codes' that work - not faith.
If every one behaved *as if* they were being observed, there would be no need for CCTV cameras! Children can behave that way because they have the 'scaffolding' of faith in a god - adults because they see from experience that it make society a better place.