Hello. I attended private school in Connecticut. I wouldn't make too much out of the fact that public school teachers have to have certain requirements and certificates. All my teachers were beyond competent in their subjects. Where I think the true benefit exists is in the student/teacher ratio. My school was 11 teachers for every student. That's a very, very low ratio that means your child will get a LOT of individual attention. Some other perks are that you don't have to attend school 180/Year. You get roughly twenty fewer days depending on the school. And, if you're Jewish (like I am) you get all of the Jewish holidays off from school so you don't have to miss school and make up a bunch of homework just because you went to shul on Yom Kippur. I would say another benefit is that your child will be extremely well-prepared for college as that is the goal of a preparatory institute. We also read a much higher level of literature than the public school near us. We also had Latin as a mandatory class (which I do think really helped me) and a daily assembly that featured a guest speaker, musician, artist, dancer, athlete, etc. That was very unique and extremely beneficial as an exposure to different cultures, artistic styles and prospective careers.

That said, my three best friends (and a 4th from Finland so we won't count his eduation here) all attended public school and we all met through arts. One is a professor of poetry, one is a librarian and one is a restaurateur. So I don't think anything played into their intelligence, love of art, culture, etc. aside from their personality and their families. Meaning, what goes on at home will likely have more of an effect on your child's future than the school they are in (provided you're in a good system which, since we are in a Connecticut forum, I will assume we all are).

My wife is from Europe so my daughter's first language is going to not be English. She will probably learn two languages before English becomes a major factor for her. I'm hoping that our house will provide enough of a base for her that public school will be an option because, as you might have noticed, private schools are vomit-inducingly expensive right now. So we are absolutely going to start out in our public system and see how it goes. We might move to a different town for high school just to get the "best" public high school we can for our daughter.

tl;dr: Despite attending private school myself I will be starting my daughter in public school and hoping she can remain in that system for as long as possible.