If it was just about salary and benefits, then one would expect lower-paying private schools to report lower teacher satisfaction. But that's not what we see:
One of the things that I realized over the years is that those kinds of comparisons are flawed in one important respect. They include salary and health benefits, but they don't include other perks (some of which may be off the books, and are almost never made public, but all of which are separate from traditional benefits and usually not publicized). The most significant one is free tuition for staff members who have students enrolled in the school. How much of a perk is that? Well, in California, the average private high school tuition is $19,830. Elementary school is typically lower. But the amount varies by state and by school. The most prestigious school in my area charges $52,500 (plus a one-time registration fee and a few thousand more to cover books, activities, and meals). Start talking boarding schools, and you could be looking at as much as $89,000 (though it may be that the teacher perk wouldn't include living expenses in such a case).
Nor is that necessarily the only perk. For instance, the most prestigious school in my area offers free meals to faculty members. That may really only mean lunch, depending on the teacher's schedule. But even free lunch for the whole school year is a considerable savings. Also keep in mind that teachers are typically given free room and board at boarding schools, and there is at least one that offers retired teacher housing as a perk. Think about how much savings we're talking about in school-year room and board.
Because these arrangements are typically not public, it's impossible to do an accurate assessment of how much they affect real teacher compensation. But it does not take a mathematical genius to realize that even an average amount of tuition included as part of a teacher compensation package would materially affect the comparison.
Religious private schools tend to be cheaper by a considerable amount. For instance, the range for Catholic schools is more like $4,000 to $10,000 (not including discounts for parishioners and siblings), up from earlier generations but much more reasonable than many independent private schools. This affordability is partly due to donations and subsidies from the larger church organization. However, Catholic schools can pay less in part because, though the supply of free monks and nuns has fallen dramatically, there is still a pool of devout Catholic laypersons who can be called upon. 97.4% are lay teachers now. Many of them are graduates of the schools in which they teach or of other Catholic schools and see their teaching as part of their service to the church. In other words, a disproportionate number of them are of the "saint" personality type described earlier. Even so, there is turnover, particularly among unmarried women (a smaller demographic than in public schools). In other words, people who actually have to live on their salaries will tend to move on at some point. One of my former colleagues left Catholic school teaching in part because she did the math and realized that she was never going to be able to achieve her financial goals. In any case, the situation is substantially different from that in public education or even in other private schools.
Here's one more wrinkle in the public school/private school comparison. People who qualify for a full pension may retire from public school and continue teaching in private school. At least in California, more than a certain amount of public school teaching leads to a proportionate decrease in the amount of the pension payments (so people can't be getting pension payments and a full-time salary concurrently). But private school teaching isn't counted against the retiree's pension. In other words, someone could get full retirement benefits plus a full-time private school salary, and the total would be considerably higher than the former public school income even though the private school salary, taken by itself, is not as high. I can't find stats on this, but I've known several people who did it. One even became headmaster of the private school in question. That's just one more way in which private schools can get good teachers without having to offer competitive salaries.
As far as autonomy is concerned, yes, that's important as well, and I'd like to see public schools give teachers more of a role in decision-making. That's another important reform. I didn't mention it before because I was trying to keep my response shorter. (Obviously, I failed at that, anyway.

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Do those proponents want to improve public schools? Actually improve them, I mean, and not just complain about salaries and funding? Then they should stop the endless and tiresome "more money for teachers" demands and start addressing the actual structural problems. Those problems will never be fixed, of course, because the Iron Law of Bureaucracy renders the status quo as implacable as iron, but it's what must happen as a prerequisite for substantial and permanent improvement. Jaime Escalante bucked the system for as long as he could with remarkable success, but he got threats and hate mail for it, and the system beat him in the end.
The iron law of bureaucracy seems like a good excuse for some people (present company excepted) to do nothing. In any case, if you compare schools today to schools decades ago, it's clear that a lot has, in fact, changed over the years. It's important to remember that a lot of decision-making is still local, and much of that happens in small districts. In most of those, we could count the number of bureaucrats with any real role in decision-making on our fingers. Someone who wanted to have a sit-down with any of them could get one within a few days, at most. Someone who wanted to talk to a school board member could probably set something up in the same amount of time. In other words, none of the people in charge are inaccessible.
Of course, one meeting may not do the job. School board meetings happen once every two weeks, and anyone who wants to speak can (for a maximum of three minutes). Have something longer to say? Divide up the material and recruit friends to deliver it.
The key is being willing to persist and having a decent approach. I've known people who generally interacted by insulting people or shouting at them. Shockingly, they didn't get anywhere. But if one is diplomatic and plays the long game, real change is possible. That doesn't mean it always happens--but it can happen.
I was fairly outspoken in the school district where I spent most of my career. (I know, shocker!) But in the beginning, I very seldom got results. (A common joke at the time was, "If you want to kill a proposal, have Bill speak in favor of it at the board meeting."

) However, I kept going, and over the years, I managed to effect at least some change. I reorganized the competitive speaking program and rewrote the curriculum for the revised courses. I successfully advocated changes in the graduation requirements, got teachers more of a role in decision-making, created an interdisciplinary combination of AP US History and AP Language and Composition, improved access to honors and AP classes, successfully advocated for the rolling back of some undesirable contract provisions, advocated successfully for numerous changes in the English curriculum, expanded the use of technology, and many other things. I also headed off a number of undesirable changes.
To be clear, the list of times I failed to effect change was longer. And I was by no means a solo act. I contributed, but a lot of other people did as well. Also, there were teachers, community members, and even students who successfully promoted other changes in which I wasn't involved. The point is, though, that change is possible.
This is especially true in a time in which school boards meetings are televised and/or left for posterity on YouTube, but the direct, personal touch is always best. Congressmen may be able to avoid local town halls, but school board members can't avoid their own public meetings. And there's nothing like a standing-room only crowd to get a politician to refigure the political calculus.
I've seen administrators fall. I've seen school board members defeated for reelection. I've also seen both groups shift position right before my very eyes. As for the iron law of bureaucracy, remember that high carbon levels make iron brittle, and low carbon levels make it bendable. Either way, it can be fractured or bent.
Jaime Escalante had the misfortune to have been fighting a much larger bureaucracy and in having large been a solo act, or at most, a duo. In a smaller district and/or one with parents groups able to become more involved (tough in lower-income areas owing to people needing to work long hours, lack of affordable child care, etc.), the outcome would have been different. Even as it was, he succeeded for a time and benefitted a lot of students in the process.