I think your experience with the public or private schools will change largely on your child and your level of involvement in their education.

For example: I'm one of three and we all went to private schools in the area for high school (except the youngest who got home schooled the last two years). I didn't like private school socially because it was too small but I do think it gave me a step up academically when I got to college. My middle sibling needed more individual attention so private school was the best option for him. My youngest sibling got bullied pretty badly (they're trans) and so then got homeschooled. They thought about public school for the last two years, but decided against it and instead they were tutored through an online high school curriculum.

The three of us are quite different and I think I probably would've liked public school better and made more friends, but my younger siblings probably needed private school.