This dead horse has almost been beaten to eternal death. I will agree with all the pros and cons listed here, and how important personal choice is. I went to Catholic grade school, public high school and then Marquette. I loved public high school because of the variety of learning, clubs, people. My parents made the choice mine.
When my kids came of high school age, the choice was theirs as well. However, my kids went to public grade and junior high and then Catholic and private colleges. My youngest had a learning disability, a problem with decoding. It took a few years to work through the public paperwork and testing and their conclusion was "there are other kids worse" or "she is just young for her grade" as funds are limited for tutors. So, we took her to Huntington Learning five days a week and all summer where they diagnosed her issue in 30 minutes and put together a learning plan to overcome it. We are talking about a kid who scored in the top 1% in state tests that were multiple choice (where she taught herself to cheat)...could whip through math fact cards, knew sentence structure singularly...but if you gave her a math word problem or asked her to write a simple paragraph simply couldn't, she couldn't read a beginning reader book but only could pick out words....it would be blank for her. We were pleased with c's.
I know there are other kids worse, and public funds are limited, so we chose the private tutor route with Huntington...which cost a lot more than a private grade school. If she was in a private grade school, the pressure would have been too much and she would not have received the attention. She could float at her own pace with kids her level in public school as she worked through her issues after school for many years.
Come high school choice: her older sister was all AP everything, great athlete, wanted the Catholic high school, college prep, very competitive experience. The younger daughter picks the school her sister went to. With caution, we gave her her choice, although the first semester was an emotional struggle despite being placed in lower levels. But her grades came and she was at scores of like 120% at that level. We went to the principal and president, explained her issues, and asked to push her up levels mid-semester to challenge her as she had caught up with seven years of hard work. They agreed, and she continued to flourish to the point where she outscored her AP Honor student sister on the ACT including a 35 on reading, was a multiple year academic All American in her sport, won many other scholarship awards and continues to do so. No way does she get moved up in a public school. Why? Because I was paying and had that voice. The public school principal most likely would never have even met with us.
Very long story short: there is no one right answer for parents...and those right answers may be different as they advance tomorrow.
Funny ending at her graduation as she had all her academic awards in front of her...I told her that I most most proud that she worked so hard to overcome her learning disability. She said with teenager disdain, "Dad, what are you talking about, I never had a learning disability".