Every single country that's kicking our ass in education (which is a lot of them) is doing so with a public system.
The common threads tend to be leveling the playing field so every kid gets a top quality primary education (which would not happen under a privatized system, hence why they are not used). Requiring a much higher level of teacher education and training, right now in most four year universities, you go to the school of education as a back-up plan to other graduate programs. In most other countries, getting into a school of education is like trying to become an MD. Teachers in most of them tend to be unionized as well, but haven't collectively bargained to the level they have here.
The differences I'm sure both of us would agree with that are employed elsewhere are pay incentives and a tenure system that's much closer to the university system. The big ones IMO are a much lower level of stress on standardized testing and the increased ability of teachers and faculty to do what they feel it takes to get every kid up to par.
Both our current educational model and a fully privatized system (even with government vouchers) have a common problem: public models have a tracking system that was built for a time when most Americans weren't going to college, so it focused most of its work on getting a very-well educated small group of people, and most others on out into the work force. It worked for the post-WWII industrial boom. Any privatized system will have the same thing occurring through inherent natural inequality. Neither addresses the problem of why we get destroyed: Every other developed country provides a top-notch education system to every single one of its kids.
You're right about the difference in country sizes, as well. I think we would be best served by having the federal government lay a very basic framework for educational standards then have each state institute its own equivalent of am ministry of education that would be more or less autonomous from there on down.
