prh47bridge faith schools prioritise their own (whether they are CofE or Catholic) and most of them are massively oversubscribed, so effectively they turn away those not of their own faith.
You are not entirely correct.
RC schools prioritise their own absolutely. If they are heavily oversubscribed those not of the faith won't get places. In some areas RC schools are not oversubscribed with Catholics, in many areas they are.
CofE schools may prioritise their own (although many don't) but an increasing proportion allocate a some of their places on a non-faith basis, so, even if the school is massively oversubscribed (which some aren't), children who are not of the faith will get places.
This is shown very clearly in the statistics. 99.8% of places at RC schools are subject to prioritisation on religious grounds. For CofE schools that are in control of their own admissions policy, 68% of places are subject to such prioritisation.
VA schools make up less than half of the total number of faith schools
But they are the ones we are mainly talking about. For VC schools the LA is the admission authority, so the LA gets to decide whether or not to prioritise children on faith grounds. Around 75% of LAs do not prioritise on faith grounds for VC schools.
there is state funding with the voluntary aided schools possibly benefitting by 10% input from churches
Stick to VA schools as those are the ones we are primarily talking about. VA schools have to provide 10% of the funding for any capital works. Historically they had to provide more of the funding for capital works. They don't have to provide funding for running costs but a significant proportion do top up the funding they receive from the state.
Teachers are the most essential part of an education, not the buildings
Whilst that is true, if you stopped all faith schools from prioritising on faith grounds you would have to find land and buildings to replace all the Catholic schools, probably the schools of minor faiths and maybe some CofE schools. That would cost hundreds of billions of pounds. Whilst some countries do teach in primitive conditions, people in the UK expect more.