The rational answer is that people of faith want to educate their children in schools where the values are consistent with their faith.
All schools educate to a set of values - they could be human rights values, they could be humanist values, Islamic or Christian values etc. They may even look similar - ‘kindness’ ‘respect’ ‘social justice’ but they will come from a different place. So for example the pp who berated me that ‘loving your neighbour’ wasn’t a Christian value, whilst you don’t need to be a Christian to ‘love your neighbour’, or to engage with the idea, the initial theological understanding of agape comes from an entirely Christian source. Jesus himself actually.
There isn’t (or shouldn’t) be any secrecy or ‘less palatable practises’ as they are all subject to Ofsted and to teach British values and must adhere to safeguarding at the highest level not to mention the Equality Act and all the other educational legal requirements.
And the reason they can is that, contrary to popular belief, we are not a secular country. The Lords Spiritual are part of democratic apparatus as is the Monarch, Head of the Church. Faith groups enjoy privileges in law. And we are a richer more beautifully diverse country for it.
It’s not perfect but not everyone subscribes to sterile atheism.