TLDR: The answer isn't private, public or homeschooling. The answer is parents prioritizing their children's education, being involved to know when their child isn't reaching their potential, and advocating for their needs.
Here is my sincere story. I hope it helps.
All of my kids when to a private Catholic school in a small town. The school only had pre-school thru sixth grade, so seventh thru their senior year of high school was at the local public school.
We thought about homeschooling, but I was working two jobs and my wife didn't think she would be a good teacher. So, we chose the private school because we are Catholic and it had a reputation for producing the top academic performing students at the public school. We were fortunate to have that as a qualitative measurement of the results.
For my first two children the private school was excellent and they were fortunate to both be taught by women and Sisters that had years of experience. They excelled. When they went to public school they were bored because they were about two years ahead. They didn't feel like they were challenged until high school and even then, the work was pretty easy for them.
My daughter, the oldest, is a hard worker and graduated in the fifth out of around 130 kids. My son, the middle child, was probably smarter, but didn't feel the motivation to compete for grades and was more impacted by Covid. He ended up somewhere around 15th out of 135 kids. Both are in college and doing well, but my daughter didn't apply to A&M and my son was not accepted for agricultural engineering and chose another engineering school.
I am happy with how it went for them. I wished the local high school was more rigorous, but they are forced to spend a lot of resources on ESL students. It is what we have and we had to make it work.
I think the biggest failure of their education is that they are not good at writing and grammar rules. They were not encouraged to read, expand their vocabulary, or study literature for assignments. Comparatively, my high school that was 1/10th the size of theirs did a better job of having me ready as I easily passed the CLEP test and didn't have to take English at A&M.
The bad for them is that they had to meet a lot of new kids when they moved over to the public school. They were sheltered and we had to really work at parenting the first few years as they were exposed to more bad behavior than they had experience in private school. My kids had to meet talked to about cursing, vaping, pornography, and even teenage sex because they were being exposed to it by other kids at the public school. It just meant we had to be more diligent as parents and watch closely who they spent time with as friends. And I will admit there were kids we told them they could not spend time with neither inside or outside of school.
Sports are a big deal to a lot of parents. I played every sport my school offered growing up. My kids were definitely behind on team sports, which didn't bother us much. I was never going to prioritize sports over education. I figured if my kids wanted it bad enough, they would work to make it happen and I focused on their academics and the other education. Both had a lot of extracurricular leadership and community service and received generous scholarships.
My youngest was a different story. We realized by first grade he was falling behind with reading and tried as parents to correct that at home. One day in second grade, I realized the smarty was not reading all the words but instead was making up what he couldn't read by looking at the pictures.
We met with the principal and his teacher at the private school and they didn't know what to do. They suggested testing him for dyslexia, but a specialist from the public school had to come do it. It was a little bit of an ordeal.
His testing came back that he had dyslexia and dysgraphia. He was 98th percentile auditory learning. The private school had no way of helping him and it was suggested that we put him in public school. He was placed under Section 504 at the public school as learning disabled. Once a week he would be taken out of class and placed in a small group for an hour of "intervention". We spent a lot of time as parents on his self esteem as he struggled to keep up with classmates.
Switching from private to public was the right thing to do, but because he was so far behind other kids, we decided it wasn't enough. We were fortunate to find a local tutor that used the same methods and system of decoding as the public school. We hired her to do an extra two hours of "intervention" every week. He did that from 2nd grade thru 6th grade. It wasn't cheap much it made a huge difference.
The youngest is in high school now. He still has his Section 504 accommodations that will go with him to college. He has learned to not be ashamed and advocate for himself when teachers won't follow his accommodations, despite it being federal law. I have sympathy for how this makes the job harder for the teachers, but it has made all the difference for him.
He took his ACT and scored a 28 on the first try, but had low marks on the English portion with a 17. He will probably rank right at 20th-25th percentile in GPA. He is considering colleges and we visited A&M, which has great resources for kids on Section 504. Fingers crossed this summer and next fall as he submits his application. He is looking at finance, ag economics and agriculture engineering.
What we learned is that most schools are going to have strengths and weaknesses. We felt homeschooling wouldn't work for us and I have seen parents that are letting their kids fail because of their lack of hubris. The most important thing is making education a priority in the family, being involved enough to know when your kids need help, and advocating for their needs. Not everyone is lucky enough to have options and you have to make tough choices about what is important and necessary for your child.
Best of luck!