Except one thing - even if you're in private school you have to pay for both. For public via taxes and private from your own pocket. It sound more like double tax for smart kids - knowledge for rich only. And just to remind that school price can be on par with University.
The problem here is economic disparity. If people can't afford to send their children to the school of their choosing, then it's a signal wages are too low. This also applies to public schools, you can be the smartest kid in certain inner cities, and you're unlikely to do well compared to an average kid in a higher income area.
But even if we have unlimited funding for lower education, what's the purpose? The vast majority of society isn't fit for a traditional university education. And even if they are, the vast majority of university students go on to achieve what, exactly?
Sure, some kids might not be able to achieve their dreams, welcome to the club. Is the dream of a life being a scientist or some such any better than the dream of being a professional skier or race car driver? We're not robots, we're people. The obsession with academic excellence has to end.
Arguably the point of public school was to develop all bright kids, not just the rich ones. Providing everyone a basic level of education is a very different goal, and I question whether a college-oriented curriculum is good for educating the masses.
The fact that my parents had to pay for both the public school and some tuition for private school (I had financial aid) left a really bad taste in my mouth. On top of that, full tuition at the private school was less than the per-student cost of public school! To me, that is ridiculous.
I don't think that's particularly fair. We all pay into the basic social programs. I don't have children but I'm still paying into the public system and I don't believe I'm being unfairly taxed. To be able to afford to pay into the private school is a luxury.
That is always the struggle, isn't it? You're granted a certain level of education/health care/security/fire services/road maintenance via taxes, but in order to get services in addition to that you need to spend your own money.