Thanks for this research example; I'll definitely use this in my Research Methods course. Even with the lottery based studies, this is quasi-experimental research, so we should not make strong claims about causality here. --A much more compelling research design: randomly assign children at a young age to charter school or public school and track their academic performance over a number of years. This experimental study would provide much clearer evidence about the relative effectiveness of charter schools and public schools. And it's not that far-fetched; similar experimental studies to assess the effects of preschool programs were conducted in the 60s and 70s.
For those with ethical concerns about conducting an experiment like this: given the fact that we don't have clear evidence about the relative effectiveness of charter schools and public schools, we are not knowingly exposing children to any substandard learning environments. In fact, it is only by conducting rigorous experimental studies like this that we can determine whether or not charter schools "work" better than public schools.