Directing money to a charity, to avoid the money going to the government, is explicitly a way of directing your money away from the common good of the public (which is after all theoretically what the government would spend it on) and towards the specific spending goals which the donor supports. The idea that a wealthy person has a 'better' idea of how to spend their money than the collective will of the people expressed through government is fundamentally antidemocratic.
A billionaire spending money through a charity that promotes a particular kind of schooling, rather than paying the money as taxes into the general fund where it would be used to fund education as directed by a democratically elected government, is basically a way of using their wealth to obtain a bigger-than-one-vote say in how education spending works, and to bypass democratic decisionmaking about how to spend money for the common good.
Of course, democracies are imperfect decisionmakers and they may not always reflect the spending priorities that will maximise collective good - but at least as institutions they are intended to balance interests, and provide a measure of fairness and equality in how decisions are made. And why would we believe that individual wealthy charity donors will make better decisions on how to direct spending?
An individual billionaire deciding that all schools should work the way they would prefer might be a positive way to end-run around institutional inertia and advance the state of schooling for everyone; or they might just pay to have every school library stocked with books about how humans coexisted with dinosaurs. They're one person, unconstrained by how the rest of society would like to improve schools.