Charter schools ARE public schools. People would not get vouchers for them because they are free. There is a selection process though in theory they are open to anyone. Selection process can be based on location, grades, skill set in certain areas, etc. I went to a magnet high school. It was great, but it has nothing to do with this thread except that in academic competitions we beat the best private schools.
You are correct, though (at least in Florida) there is also a difference between magnet schools and charter schools. Magnet schools are regular public schools schools that have been redesigned, either in whole or part, around a specialized curriculum - arts, science, international studies, etc. And you are correct - the academic outcomes usually exceed that of private schools.
I think there are two reasons for this. First, as you noted, the magnet programs are competitive to get in to. Unlike most private schools, parents can't buy their way in - the student is accepted on merit. The second reason, in my opinion, is that the teachers and administration are still held to the educational levels of a public school, and have the benefit of a union to keep working conditions tolerable.
Private schools are only as good as the for-profit model they choose to operate by. Private schools can often be the best of the best, or it can be Trump U at the lower grade levels. They are also unconstrained by teacher's unions, so those teachers are often paid less and have less training/experience/credentials. Charter schools are private schools taking public money.
SLD is 100% correct that the Republican version of "school vouchers" is really a means for the religious right to get public funds for their religious schools. In Florida, almost 70% of the students who got vouchers in 2015 used them for private religious schools. And as Elizabeth Warren points out in the excerpt Trausti tried (and failed) to play "gotcha" with, the Republican-backed voucher programs don't really help students of lower income people because it doesn't allow for true school choice. Because it is a "tax credit" system rather than a true voucher system, it doesn't really help the neediest students, so they end up being left behind in a failing school that now has even fewer resources.
A true voucher program, such as the one Elizabeth Warren suggests, would still allow parents to choose religious schools unfortunately but would otherwise be more equitable. Florida currently spends $7,105 per student (one of the lowest in the country). Those funds, however, are not currently distributed dollar for dollar to each of the schools in a district or throughout the state. Wealthier neighborhoods pay higher property taxes, and thus get a larger budget for their neighborhood schools. Palm Beach County spends $9,179 per student, while Broward County spends $8,117 and Miami-Dade spends $8,725. Palm Beach County is some of the most expensive real estate in the state. And within each county, a similar disparity plays out between the wealthier neighborhoods vs poorer ones.
Warren's idea is to have the money split equally between the students, and have it follow the student but paid directly to the school. As such, parents would be able to have their children attend neighborhood schools because those schools would be getting the same funding per student as the wealthy neighborhoods. Likewise, parents could choose to have their children attend schools closer to their work if that is more convenient.
