Some contributors seem to think the public sector cannot add to national output or incomes. This is not true in either accounting or real terms.
A person attending a private school is paid for out of fees. Their education adds to national output. A person attending a state school has a similar education but their parents do not pay directly. It has a similar effect on national output to the private school place. The state provision is as much output as the private sector and is paid for out of taxes.
There could be a productivity effect. If in a particular activity public sector or private sector productivity lagged then the total economic impact would be affected accordingly. In the case of schools private schools may well have higher staff ratios to state schools.
When the UK had a lot of nationalised monopolies producing energy and transport we had an efficiency problem in those sectors. In those cases privatisation led to an improvement in productivity, which was possible to bring about because people accepted the principle of paying for use out of their own incomes. Market pressures encouraged adoption of better technology and more efficiency.