Pointless -- "atheist" schools already exist. All non-faith schools are "atheist".
Lavender my observation about "community" is not with regard to the parents of children at the school; it's with regard to the support (some) faith schools receive from the community of the church with which the school is affiliated. Not all in fact not most of the members or attendees of the churches I know that have affiliated schools are parents or would-be parents of school children. They are individuals who have faith or wish to have faith -- and/or who enjoy the music, ritual and support found in church services. (As an aside, I do find your language unnecessarily hostile and offensive. I'm sure no one on this thread "feeds their children crap".)
My question was and still is what is it about "faith schools" (in particular CoE schools) that makes them so "good"? My hypothesis was and still is that perhaps the schools are good because they benefit from the support they receive from the local church -- church members (not parents) who devote time to help at assemblies, to read with children, to help with school fairs and celebrations, etc. Perhaps if that level of community support were replicated in non-faith schools (as it undoubtedly is in many cases!), there would not be such a divide, or perception of divide, in the quality of faith and non-faith schools. It could also be that faith-schools put a stronger emphasis on moral education and discipline, both of which are conducive to good education. It is important to think carefully about what distinguishes faith schools from non-faith schools because if you seek to eliminate them without first identifying and replicating what makes them "good", the overall quality of education in this country is likely to suffer. (The Muslim families who I mentioned above as parents of children at my local CoE school, chose the school because it is good and actively encourage and support their children's participation in the "C" aspects of the school because they appreciate the moral teachings, the discipline and, yes, the emphasis on faith).
To the broader question of whether state schools that are also "faith" schools should be required to have an open admissions policy: If faith schools were "open", would it be on the understanding that parents who send their children to the school, regardless of their own faith, are happy for the children to receive a faith-based education? Or would the movement seek effectively to terminate faith schools altogether? If the latter, then there would, of course, be cost implications for the state presumably the state would need to acquire the church-owned buildings or pay rent for them and would have to absorb the costs of the financial contributions currently made by the church (whatever that is) and somehow absorb the pastoral care roles currently provided by the church - none of which is impossible, but those who advocate the termination of faith schools should think carefully about all the implications.