That's not always the case. A lot of private schools rely on the public school system to fill in for remedial classes that they don't want to staff for because the cost would be too high, so the public school sends a bus over to pick up the kids at the private school that can't read or are behavioral problems and takes them to the appropriate public facility. So the private school takes your money and then instead of using it to teach your special needs kid, sends them to public school for tutoring.

Case in point, one of my nieces needed speech therapy when she was attending a private elementary school, because her mother thought the private school was better. Well everyday after she was in classes she had to leave her private school and take a public school bus to one of the public schools that had a teacher trained in speech therapy because the much better private school didn't have anyone on staff capable of teaching this. She was not alone on the bus either.