It's interesting to look at the history of various debates that raged in society and how the various pundits at the time thought things would work. In this case, at least, the Catholic Church was completely correct, our education system has become exactly what they predicted it would become.
It was really a no-brainer.
Government has no business doing anything but protecting us from other countries and each other.
The Protestant arguments were fraudulent because their whole purpose was anti-Catholic, anti-Jew etc. and seeking to impose a Protestant religious view on everyone through government. Most people were and are bigots of one sort or another looking to use government to indulge them, something not possible before about the 1850s.
Everybody wants to be Constantine. Government MUST be made too small for it. Bigots must not be able to use the state.
I recently learned that a big Andrew Carnegie foundation decided to fund pensions for many Protestant college and university professors, on condition that their schools adopt a "non-sectarian" policy.