I agree with your opinion that "Parental involvement is key in making sure the child gets what they need from the public or private system"
However, I'd like to address your point that "A private school is no guarantee that the education or environment will be better for every student"
If I study for a test, there is no guarantee that I will do better than if I were to not have studied for it. Nevertheless, basic logic and evidence dictates that for the vast majority of tests that I take in my life, if I study beforehand I will do better than if I were not to have studied.
Private schools for the most part, produce vastly better standardized test scores, are safer, provide a more controlled learning environment, and are generally better at preparing kids for college. The statistics really don't lie here. While it's true that maybe a private school isn't best for everyone, let's be reasonable here - all things equal, private schooling is a better option if you can afford it.