Not on average. See above.
That was the case back in our old city. Students in private schools do better because the parents have the money to pay for private tutors on the side. The majority of my wife's clients were private school students whose parents were shelling out $2k+ a month for each kid on tutors alone (times 4 kids). Oh, and private schools teach the exams in the last year.
Private schools do better for a variety of reasons. (I think it's a bit silly to act like they're all getting tutored by public-school teachers though.) I don't think it's necessary to go into those, but the bottom line is that they do at least as well as public schools with teachers making less.
They certainly have a value to the club they play for, which is why they get paid what they do.
Did you take microeconomics in college? I'm not trying to be mean but what you're saying doesn't fit with basic economic theory. If people shat out gold, gold would not be worth what it is today. Think about food. It's the most important thing in the universe to humans, but it's cheap because the supply is plentiful and food is fungible. One donut is the same as another donut. Basketball players get paid so much because they are the best in the world. There are only a couple hundred of spots in the NBA. Sure a lot of people play basketball but Kobe Bryant is not the same as some high school goofball.
Supply and demand is more complicated in this case than you have considered as you haven't taken pricing into account. Pricing is artificially set by the fact that government is involved and sets a price ceiling and essentially a price floor.
No, as we've seen now if anything the unions set the price higher and the private market shows the natural wage would be lower.
Without government involvement at all in education you'd have a rockstar school with rockstar teachers all making $500k and teaching billionaires' kids.
Nothing is stopping billionaires from doing that. That's the situation as it exists today. And yet teachers at elite private schools and even elite universities don't make THAT much. Because many people can be a teacher, even a good one.
You seem to think elementary school teachers are just glorified babysitters. If that's the attitude you have, that's what you'll get, and the kids in your schools would come out at an appropriate level. Refrigerator privileges anyone?
Not at all. I just don't think that teaching requires that raw brain power and massive education that you do. I imagine that things like patience, kindness and fairness make a huge difference at that level. Being able to do differential equations is irrelevant. Having a PhD is irrelevant.
Don't get me wrong, I respect teachers and I think they're more important most of the people on Wall St. But they don't deserve to be paid above-average salaries by the tax-payer who in unpleasant environments where they can lose their job at the drop of a hat and have less and less benefits.