So, since things vary so much from place to place wrt church schools and their relationships with the local communities, what about this scenario:
St B's is an outstanding C of E church school in a village. Other nearby schools are non-church, and easily driveable, although not quite as successful. St B's has church people going in to school weekly to hear children read, lead assemblies, and do music. It has church people on the governing body. Children go to services in the church a couple of times a term, and also for lessons from the vicar, who is a former teacher. Its application process prioritises local people and the church attendance criterion is hardly ever used.
A local campaign is mounted to tell the church to get out of its school, and after consultation, the church goes. This is sad but actually, the church in the village is up to its ears in buildings work that needs to be done on the Grade I listed building and its elderly congregation barely scrape enough money to cover the running costs, let alone anything on top. Financially, the church is in crisis, so in a way, the school telling it to butt out has a silver lining. The school had a generous loan from the church some years ago - around half a million - when the church was richer, and the church owns the land on which the school stands. The 'divorce settlement' between the church and school means that someone - the local council which is also massively in debt - has to buy the school land from the church, and pay back the half million. The people who go in regularly as helpers or governors all know when they're not welcome, so they take up the plea from the school down the road who is only too happy to have them contributing to their school life. There aren't many other local people willing to give up their very limited free time to help at the school - everyone is either out working, not interested, or is too old and frail. With so many of the governing body gone, the school is weakened. The vicar is incredibly busy and is much in demand from other local schools, so although it is sad that he is no longer welcome in St B's (the name is changed to something non-religious, of course), he is still able to do much to help the local community.
It's silly to ask if, at the end of this 'keep faith out of my school' campaign, St B's would be better off - of course it wouldn't. Would anyone benefit from this arrangement? The parents who mounted the campaign, sort of. But their children?