I'm not sure I understand the OP's point.
If she is saying that people shouldn't participate in school systems that they don't believe in, then that is clearly impossible. I don't send my child to a faith school, but I am still participating in a system segregated by faith.
If she is saying that if people believe that certain systems give advantage to group of children over another, and that to give your child the advantage means that your opposition to the system is not credible makes sense. It would then follow that if you send your child to a faith/grammar school or take up a private school scholarship, you are participating in creating disadvantage for somebody else by taking up a place that another child, possibly from a less advantaged home, could have received. In much the same way people can't claim to be opposed to selection by postcode and then send their child to an excellent school in an affluent area if other, undersubscribed schools in poorer areas nearby are available to them.