The government does not call their schools government schools. Retaining support for their monopoly would be harder if they did. They call them public schools. This gives people the idea they are like a public golf course, privately run but open to the general public. The government does not want people thinking of schools as a branch of government like the IRS Department of Motor Vehicles.
Government schools are government agencies paid for by tax dollars.
In most cases property taxes pay for them. These property taxes turn citizens into surfs renting their homes from the government. If one fails to pay his school tax the government will come and remove him from his home by force and turn it over to another surf who will pay. One is never free from these payments no matter how long he had possession of the property. In effect, one leases his property from the government.