True, plenty of private schools, but it depends on your definition of monopoly. Microsoft had competition but that didn't stop the lawsuits from the Feds.
Government education may not be a monopoly in appearance but in many cases it sure does behave as one. Look up the examples of local government education bureaucracies doing much to attempt to eliminate those private schools as competition.
Washingto D.C. is an excellent example of this.
In many areas I would define government education as a monopoly simply because they do everything they can to prevent you from not entering your child into their system. And they use government force to do so.
Its not a perfect monopoly, but the market dominance of state education is greater than that of Microsoft, or Standard Oil, or any of the other companies we tend to think of as monopolies.
But it isn't a monopoly. There is no single entity controlling all of the public schools. There is no Gates or Rockefeller of education.
Each school is controlled by the state and local government, which is in turn controlled by elected officials. There are federal and government regulations, but aside from that, it is run by the people.
If you don't like what your town or district is doing, get on the school board and change it, or vote for someone who agrees with you. And if you really don't like what the state is doing, offer your own option through private education.
It is monopolistic in the sense that you have to pay for it whether you use it or not. Private schools may only be attended by those who can afford to pay twice and even then, the establishment caste gets at least a full cut.
If I were to set the policy, I'd subsidize students rather than teachers and administrators, etc. I.e., demonstrate learning and we'll pay you $XX,XXX to cover your costs whether tutors or books or online courses or brick and mortar schools.
Strictly speaking, government-operated schooling is an oligopoly, and one very stringently protected against meaningful competition. As the other two replies at your reply level I see here pointed out, a private business having the same market position in the United States and the same preferences that government-operated schools have would probably be illegal.
"This attitude is a natural outgrowth of our decision to operate education as a monopoly."
This is blatantly not true, in the US we do not operate education as a monopoly. There are plenty of private schools.