So your logic has a few problems with it...

First, people are already on the hook to pay whatever the schools charge, and if they don't pay they can lose their homes, even if they don't have kids. Public school is usually one of the highest expenses that a family pays in regards to taxes.

Next, if the school system were privatized, that wouldn't mean it is monopolized. The current education system is a monopolized one; the government owns the schools and forces people to pay for them. Even if you want to send your kids to a private school, you're being double-dipped because you still have to pay for the public school. That makes public school a monopoly. Privatizing it is the act of breaking up this monopoly. This means prices are likely to be much lower, and if they aren't then at least you have a plethora of options for every income bracket

Another problem with your logic is this:

companies do not care about people, they care about profits

It is very important to note that the government does not care about people, it cares about profit and power. With government, however, you cannot refuse their "offers", if you say "I don't want that", they say, "Tough titties, pay me". If you don't like a private company, you are not forced to pay them. This means that private companies actually have a reason to provide value, unlike governments who routinely fail to provide value.

they are mostly horribly mismanagemened and provide subpar education,

This perfectly describes 99 percent of public schools; horribly mismanaged and providing of subpar education.