Went to a private Catholic school too. Prayer, uniforms..the whole nine yards...and having the best looking co-eds in the city (cough). Never was a snob about it but one thing was sure: I can talk to my teachers after and before school.
The most important point is not only access to the teachers but in private or public: parents should be active in the lives of their kids. It applies to both public or private education.
More importantly most of the teachers could move over to a public school for higher pay and better benefits, but they stay because they are doing the job for a higher calling.
I remember back when I was in college people who were too dumb or too lazy to get anywhere in engineering or architecture would tell me they are switching to a humanities major so they could then get a teaching credential which was “a lot less hassle.” People like that who only see it as “a job” wind up in the public schools. I have no idea what the percentage is, but I suspect it is high.