Based on Rothbard's description... The educationists who pushed for it were not a completely homogeneous group, although they were reasonably well organized.

Some simply thought education was good and weren't bothered by the coercion.

Some subscribed to the absolutist State view going back to Luther and Calvin—people saying things like "in these schools the precepts of morality and religion should be inculcated, and habits of subordination and obedience be formed", "that a teacher must lead his students to accept the existing government", "the children belong to the State and not to their parents", etc.

And some had egalitarian aims along the lines of what you describe. "The vigorous championing of the public school's leveling role appeared again and again in the educationists' literature. Samuel Lewis particularly stressed that the common schools would take a diverse population and mold them into "one people;" Theodore Edson exulted that in such schools the good children must learn to mingle with the bad ones, as they will have to do in later life", and others are cited too.

Although you may be surprised at how far they go. Rothbard says that Owen and Wright—Wikipedia identifies them as utopian socialists—had very extreme plans involving the State raising all children away from their parents in a uniform environment, as the logical conclusion of making the childhoods of the rich and the poor be totally equal; those are the only two he cites advocating this, but he claims they were very influential, the essays published with approval by a great many newspapers, etc. Could be.

Back to what you said: The "including teaching it's boredom, authority, and pointless busywork"—well, authority yes, but the boredom and busywork seem pretty clearly not part of the original plan. I think that's an accident which has developed since then. Which brings me to this point:

Supposing that the above describes the state of affairs in the mid-1800s, a great deal of time has elapsed since then. Private companies with clear goals develop cancer (that is, pockets of people acting selfishly at the organization's expense and subverting the organization's usual methods of enforcing its objectives) on smaller timescales than that. You should expect some ... evolution.

Generally, I expect incentives to direct the evolution of an organization. If we consider the "customers" of an organization to be those who decide if it gets funding, then the customers of a public school are primarily the voters or the politicians they elect, not the students or the parents—except insofar as those can influence voters. (Also, sometimes a school is allocated funds proportional to the number of students, so students voting with their feet could be a force; on the other hand, switching schools involves hassle, losing one's entire social group, and likely a longer commute, so it's likely not used often.)

We might expect this to mean that schools would tend to implement policies that make things pleasant for school employees at the expense of students, as long as they can't be turned into public relations disasters. Continually increasing the homework load, starting school extra-early for teenagers despite decades of research about sleep schedules showing that's a bad idea, and assigning busywork (either in the sense of "literally to keep the students busy while the teacher can sit and relax", or in the sense of "make all 20 students do an extra 5 problems because of the 1-2 people who need the practice") are all examples of this. (Note that, for all these, opposition can be dismissed as "the complainers are lazy bums".) I've had a case where a teacher—a nice guy and a smart guy, I liked and respected him—told me that, if he made an exception to implement my request, he'd have 100 parents beating down his door with their own requests—so, keeping his negotiating position with parents was more important than designing a better program for his student (well, from his incentives, it was more important).

It could be worse. The book "Little Soldiers"—based on what I've read[1] about it—depicts what is apparently considered good practice in modern Chinese education. Students starting at age 3 are required to sit very still and do a delicate and pointless task over and over, and are yelled at whenever they mess up—all to establish "discipline" as early as possible. Parents are expected to fawn over the teachers and, under the table, to buy them gifts. This seems pretty clearly the end state of "optimized for school employees at the expense of students and parents" (well, I guess I could imagine it going even farther...). It seems American parents are more unruly and have a better negotiating position than that.

But generally, beware of ascribing purpose to any particular part of school. If the school was laid out in a coherent design by one person, and it's relatively young and small, and has good competition—then maybe you can expect most of it to be part of the plan. But public schools that have existed for a long time... When you experience busywork, it's not the taste of egalitarianism; it's the taste of organizational cancer.

[1] https://old.reddit.com/r/slatestarcodex/comments/cz48fp/litt... and the discussion at https://slatestarcodex.com/2020/01/22/book-review-review-lit...