I made the case above that choice in education is not exclusive to private schools, and there I perhaps worded that statement poorly and should have said "Choice is not the defining characteristic of private schools". You still have just as much choice in the type of education your child gets without private education.
The author's argument revolves around removing the choice of paying for quality, not removing choices of the type of schools and education available.
If we got rid of private schools, we'd still have all the choice, but the choices and quality would be available to everyone, not just rich families.
The authors argument revolves around removing choice so everyone gets the same thing so we are forced to make that one option higher quality. I'm not saying that is the only way to do it, but that is what this person is arguing for and that is what I'm arguing is a bad idea.