schooling in the old way

By private education you mean education you pay for, right? It sounds like you are conflating private education with religious education, which isn't right. Maybe you just mean that the Catholic school just so happens to be private, in which case you can ignore my previous statements.

Assuming you meant the definition above, private school is almost always better. The geographical area of the school is very important too - a public school in a good area may get better grades out of students than a private school in a worse area. It happens, but it's unlikely.

The main problem with private school is affording the tuition, and your son/daughter being a privileged little shit, but the pros outweigh the cons generally speaking.

there are definitely non-religious private schools, but in the US, private schools (especially in regards to primary schooling) is vastly religious affiliated, mostly Catholic.

Catholic schools usually come to the mind when you talk private schools for children here.

Ah, well it's the opposite in the UK. We have state sponsored religious schools whereas the private schools just stick to educating people mostly