Finland has much stronger teachers' union and pro-union laws than the United States, and pretty much no private schools, and at the same time has been getting stable, good results(http://en.wikipedia.org/wiki/Programme_for_International_Stu...) in international comparison studies.
You will never find a singular change that can turn around your education system and getting rid of the teachers' unions surely is not the magic pill.
Finland is a wild outlier. The gap between Finland and the rest of the European (+ Euro heritage nations, e.g. Canada, Australia) nations is huge. By comparison, the Euro-like nations below Finland all cluster together.
For unknown reasons, Finland is special. Bringing up Finland is a bit disingenuous.
You could bring up more typical Euro nations, but unfortunately several of them get pretty decent outcomes with privatized school systems (e.g., Sweden).
That's kind of his point, I think. He's saying private or public isn't the point, whatever you do is going to be more than just one buzzword-thing like "take down unions" or "voucher system".
There isn't just one thing wrong with the US education system, there are countless things. Many of which aren't unique to a public or private environment, just plain old terrible schooling.
You will never find a singular change that can turn around your education system and getting rid of the teachers' unions surely is not the magic pill.