I live in the Los Angeles School District, and our local elementary school is a charter school which accepts all students in the area. Like almost any complicated issue, the topic of charter schools should be treated not as a single entity which acts and performs identically in all instances, but as an alternative that may be better or worse in a given area at a given school. Of course there are ways to mislead with statistics and I haven't researched the area well enough to know if CREDO at Stanford is looking at performance objectively - if they are, one can quickly see that they report both improvements in some areas and some locales as well as schools that are doing worse. If you have you kids in a school that happens to be doing better than a previous public school (and in a way that is really unambiguous to you), you aren't going to appreciate being told that your school must change the way it works based on some national statistic of how well or how poorly charter schools are doing.
In my case, I was not in the area before the local school become a Charter school and I can only compare it to the school I went to 40 years ago (when a ton of things were different), but it seems pretty good to me and I'd want to be careful with changes being made to it.
With respect to common core, I have some major complaints as an engineer on the way they rolled out the math changes. They had the outline ready, but no open source materials explaining how to solve problems with the new terminology. Parents should not be forced to register with private websites to figure out how to help their kids with elementary school math. And on top of that, I found the way some math was decomposed into new ways of explaining to be more confusing, but I recognize I have learned one way and possibly they have research to show more people have a better understanding of math after learning it a different way. But in that case, I'd want to be convinced by research which I don't see being offered very often. But I probably wouldn't support eliminating common core just yet, but it is in need of some reform before it will get wider support from parents.