Private schools offer an unfair advantage to other students who do not come from a rich family or are unable to afford to go there on their own merits. It gives children the sense of privilege and the right to treat other unfairly because they are in a "supreme" school.
Private schools allow parents to have a choice in where their child goes to school. Currently, your child can have a free education but almost always only at the public school that they are zoned for. If a parent wants their child to be educated in a different environment, they should have the option to choose a different school if they want to. Some choose religious schools, schools with specialized curriculum for arts and sciences, or just a school where their student can feel more comfortable.