The only con I have with private schools, (though I have never had my kid in a "religiously affiliated" school) is the somewhat closeted atmosphere. Not snooty, or cliquey, just loads of people, though from diverse backrounds with very similar agenda. My own experience with them was my kids developed the attitude that they were somehow better than publicly schooled kids. I have had my kids in 3 different ones, and it seemed as though this attitude was fostered by the schools. Maybe it was just intense school spirit, but when we finally found a public school that could accomodate my kids, I really saw this elitist attitude blossom, and it was ugly. My kids are just now beginning to see the benefit of public schools as a "better" representation of the real world. Not everyone in public schools has aspirations of being a senator or a doctor, as seems to be the case in many private schools.
On the other hand, I would never have my kid in a school system that was not giving him/her every opportunity to become their dream, and I find that the incredibly bad public schools seem to have an agenda of their own. That is to make sure that the kids don't set their goals "too high"
Also, remember, there are plenty of opportunities for parental involvment at the public schools, so that point really is not valid, though you need to be the one to initiate the involvment, unlike private schools, where it is a necessary obligation for the school to function properly.
Good luck whatever you decide. Remember, no matter where your kids go to school, ultimately, you are their first and most important teacher. The schools work for you. Hold them accountable, wheter public or private.
Tracy