Well, this is interesting.
I am sure there are others on here who have been involved in all three sectors - public, private, home schools. For the record, we did not homeschool our children but looking back we wished we would have at least for our son. He is on the spectrum and extremely bright, like Mensa bright. I do work with some homeschool parents when they have questions.
All that said, I was a principal at a small public HS that had some of the highest test scores in the State. The State recognized us with the Governors Award, the only small school to earn it. It was not because of me, it was because of the GREAT teachers there and the standards that were established. I occasionally sub in our local public schools and I come home depressed by how the kids act and what they are NOT learning. That would be all grades but no high level specialty classes. Just look at test scores for your school and district and see what you think - it is likely very depressing. There are exceptions, but they are few and far between.
I was the principal at a K-12 Christian School and on average our kids were at least a full grade level ahead when they moved to the public schools. I loved working with those kids and they were in a very caring and structured environment. The range of parents were from barely making it to very well off. We found a way to help them keep their kids in school. Teachers worked for peanuts as I did. By this time in my life I was retired from the public sector so I could make it work. The downfall of small Privates they have many challenges, money and offering programs like sports, scholars bowl, technology, etc. It is very difficult
Lastly, after retiring once again from the K-12 Christian school I helped some parents start a Classical School. It is an awesome form of education and you should look it up if you are interested. Kids learn Latin and Greek and very high levels of thinking/reasoning and mathematics/science. Most Classical schools are smaller as the curriculum is rigorous.
My point in all of this that there are many options and parents really need to look at outcomes of previous students. I am sorry to say that I would NOT send my kids to our public schools here in town. Sure they have all of the activities, but they also have all of the problems. In many cases, it isn't the teacher who is bad, it is the system they are in. Just as everyone is free to live where they want or drive what they wish, why is it that public schools do not give parents choices, especially those schools who are failing and you are forced to send your kids there because of where your house is.
EDIT: I forgot to mention that I have also seen some very poor home schooling. Had a couple I know well bring 3 of their kids to the Christian school (didn't know they were homeschooling them) and wanted them tested, remain at the grade level they had assigned and no questions. The teachers knew where the kids were at academically and gave extra help to get them caught up but I know what would have happened had those kids gone anywhere else. Had another situation where a pastor wanted me to meet and test a kid whose mother had just moved into town. She told the pastor she had been homeschooling the kid since he was little. Memory tells me he was a freshman, big kid, 6'1" at least, 210 lbs and good looking, athletic looking kid I thought he could be a LB. Turned out the kid couldn't write even the beginning of a simple sentence. Could barely print his own name. I never knew what happened to that kid but his mother should have been brought up on neglect charges. My adopted grand kids are home schooled and they are miles ahead of public school kids using Classical Conversations Curriculum. There are always exceptions to everything.