The problem with faith based criteria is it can be used to skew the
admissions.
I live in a village. I want my child to go the village school. We're atheist, it's CofE, we'd rather it wasn't, but the community aspect outweighs that for us. Near us most of the villages have CofE schools and a couple of catholic.
People (generally naice middle-class types) in the nearest town, which has a primary geographically within a large estate, horrified at thought of their children going to nearest school, pick a village, go to the church for a couple of years, hey presto - get child into village school on the faith criteria. And then siblings. The schools round here are smaller than average and it's amazing the proportion of intake that can be siblings in a year.
People like me, who have no faith, and aren't prepared to invent one are then disadvantaged. I'm incredibly lucky, we moved to a house over the road from school before we thought about kids, so we were first once the looked after and faith criteria. Not everyone is so lucky.
If the middle-class kids in the town actually all went to their nearest school it would become a good mixed intake school they were happy to go to IYSWIM.
Something needs to break the existing system.