Great idea. I agree though that the fact that churches still own the land and provide significant funding (a new building will be about £10M. So £1 million quid to come from the church) will be a problem. Possibly start by encouraging all church schools to prioritise children who have attended any church, not just St Leafy's in the posh area of town that isn't welcoming to many - which would at least improve the social mix slightly, if only to parents who are organised enough to attend a church fortnightly for two years before applying for their child's place.
And once they realise the sky hasn't fallen in, encourage them to introduce some places open to all, so they better represent their local community.
I believe the few schools of other religions are rarely oversubscribed, but same would apply. Someone pointed out the other day that Jewish schools have practical benefits for pupils being able to provide kosher cooked food and ensure the school day ends so pupils can get home by bus etc before sunset in winter, but there's no reason other schools couldn't support packed lunches or buy in boxed meals, and tweak their Friday timetable, if necessary.