The problem under most voucher proposals is that parents could add money to the voucher. They could buy the most expensive education they wanted. Let's say a voucher was worth five thousand dollars a child. Some affluent people would add ten thousand dollars and send their kids to the fanciest private school. You'd lose the mixing by social class.
Education has basically two major missions.
One is to educate the kids. The second is to socialize them. Kids get to know kids who aren't like them, kids from the other side of the tracks, kids with different skin color, kids whose parents went to college or didn't go to college, kids who talk differently than they do. You'd lose that under a pure voucher system. Just look at suburban schools compared to inner-city or rural schools.