Some things taxed as luxuries are unnecessary (eg watches, phones), others may have a cheaper alternative but not free (eg some foods, adult clothes).
Some things which aren't taxed as luxuries are unnecessary (eg flights), others may have a cheaper alternative but not free (other foods, designer children's clothes, mortgages), others have a state-funded alternative (healthcare).
There isn't a consistent argument that whether there's a free alternative determines whether something is a luxury which should be taxed. No consistent explanation that mortgages different to private education, or private healthcare different/similar to private education or children's designer clothes different to adult supermarket clothes. It's a political choice.
All these justifications about why private education should have VAT are just that - made up to justify something people have decided on for a different reason.
You say that private school parents are not giving money back by not using a state school. But you can't deny it will cost the state more if a family decides to put their kids into state school than if they don't.
From purely utilitarian point of view, the state will have less money to spend (whether that's on the kids who are already in state school or on training extra nurses) if they pay to educate all UK children instead of only 93% of them.
this report outlines why 10% is the tipping point where the extra cost completely offsets the extra VAT. Any more, and this policy leaves the government with less money.