My ds is about to leave state primary and start private school. I'm a disabled single parent and ds school fees paid for by a generous school bursary and a great scholarship from a scientific charity because of his understanding of the subject. He's not especially gifted though.
To those trying to find a way to send their children to PS it is possible. Keep trying. Armed forces and trade organizations can often help. Large industrialists have educational foundations.
Apply and hope. Even a 15% discount from any source could be enough.