The Blairs chose the Roman Catholic London Oratory school for their sons. Proportion has been falling The Church of England and the Roman Catholic Church currently run around one quarter of all secondary schools. For the past century the proportion of church-run schools has been falling as education became a state, rather than church, responsibility. Few would deny the excellence of many church schools. Many would also accept the right of groups to run their own schools.
More church schools runs the risk of religious polarisation.
Look at Northern Ireland where the vast majority of schools are run by the churches. Living separately, and being educated apart, does not facilitate better understanding between communities. Denominational schools do not exist within the taxpayer-funded system and no religious worship is allowed in public schools. There is also a risk of social polarisation. While many church schools serve deprived communities, and educate children from other faiths or no faith at all, this is not always so.