We're a lot better off letting the free market do its work. Wealth was basically stagnant prior to 1800. The free market / industrialization arrives, and suddenly wealth began to grow. People today are, on average, many times wealthier than those who lived before.
Most people I know could be a teacher. Maybe not a good one, but they could do it in a pinch. I don't personally know a single person with the youth and athletic ability to be a professional athlete. Supply and demand.
Further, the school system is essentially socialized education, while professional sports much more subject to free market principles. The cream rises to the top, and is well compensated for doing so in the sports world. How does a teacher rise to the top of his/her profession? That same teacher is still going to teach a limited number of classes a day, and will have a limited number of students per class. About the only way to get that teacher better compensated for those abilities is to introduce the free market to the classroom. Rich people will pay a lot of money to ensure their kids get a good education. But teachers don't like that idea because that means that poor children get stuck with bad teachers who can't get a job anywhere else - the rich get richer and all that. Fair enough. I admire the principle. But it's the principle that is costing great teachers lots of money, and preventing people who could be great teachers from getting into that line of work.