WELP this is gonna be fun, let me start off by saying that I myself am a bit biased because I am the son of a teacher, my dad taught 8th grade English for a while but was just moved to the 6th grade special education department due to the fact that he is not president of the teachers union in my town.
So as a teacher's son I was raised to certain standards and I was given insight on things that other students have not been given insight too. Most of what I'm going to say is seemingly known to a few students because of my father's position.
Onto the debate I'll start off by going off of what Valt said first
1. They both have an authoritarian structure - An authoritarian structure is just about the only way you can do things with differing classes. We see authoritarian structures in every day businesses (Boss employee) government (government official ,public) and yes in schools (staff, students) If everything was happy go lucky and everyone is equal on power, shit wouldn't get done, ESPECIALLY WITH CHILDREN. That is why there is an authoritarian structure in schools, WHO THE HELL WANTS TO LEARN? The teacher's job is to teach you and no child would want to be taught. I say child because as a teen the opinion starts to divide because some kids realize to value education... while others don't. That's how I see it in High school (I'm currently a freshman) I enjoy going to school because I know I will be learning fundamental skills in life and honestly, my teachers rock, why do they rock? sort answer I'm in advanced classes so we get more privileges long answer will be touched on later.
2. They both have strict dress codes - Public school dress codes are relaxed. Don't wear slutty shit and everything will be ok. I'll go into detail now. What is against school dress code is usually something that a hooker would wear on the street. With hormones running high bad ideas will circulate when you come in with a shit barely covering your tits and pants with legs so short your panties are showing. On the male side sleevless shirts are usually a no no, why? Because no one wants to smell your BO and see your pit hair, so actually it's more of a sanitary reason.
3. They both have "Negative" reinforcement - This one is half right Schools have positive and negative reinforcement, it depends on what party you ask. For example if you asked a total delinquant and an honors student on what reinforcment they recieve, the answer will differ because the honors students are given special privlages because they are trusted by the staff and they have shown dedication to learn. Late a little bit? teacher will let you slide with a warning. Your treatment is decided on how you act in school. IT'S NOT JUST GRADES!!!! I had the nicest kid in my class in 8th grade but he was flat out dumb, complete idiot, wanna know what though? He worked to improve himself, he went to extra help EVERY DAY and by the end of the year he was going into High-school A: with full advanced classes and B: as a teacher favorite. Teachers have meeting and share things about the students. If you have a good rep that will carry over HOWEVER if you have a bad rep the teachers will be cautiose with you but if you show desire to improve you get +1000 rep points, suddenly you look like a saint for trying to improve. (That was just a story of one of my own friends in fact so I speak from experience.)
4. "Walking in lines" honestly this one is pretty fucking stupid - That rule carrys on from K-4th grade, maybe 5th. It's because the teachers need to look out for the kids and make sure they get to their next class ok, once you reach 6th grade you walk yourself to your next class and you're expected to get there on time.
5. Abridged freedoms - Explained this one already. If you show you are a good student you get more privileges as a reward. Bad students can decide to become a good student any time if they try.
6. No input for decision making - Input in school is a privilege earned, once again the better of a student you are the more input you have. On the note of decision making I will tell you of a thing known as a supervisor, one who rules over the teachers. The supervisor decides on what you learn and how you learn it, the teachers just follow the orders. Hate what you learn? Blame the supervisor that's their fault not the teacher's.
7. Set times enforced for eating, walking, etc. - This is just because you need to keep an orderly schedule. Students FAR outnumber staff in schools so by giving them a schedule you can ensure that they are safe. On the note of eating this one is a very relaxed rule. Hungry because you forgot breakfast? If you kindly explain it to your teacher you will most likely get a little talk to but you will be able to eat your food. DO REALIZE that a student falling asleep in class is extremely troublesome to a teacher because they need to ensure that you are learning the material.
Welp there is my take on the school system, in short you earn your rights, the rules that are there are for your own good, and the teacher is a very common scapegoat and people don't even realize it. Have any other questions about how schools are run? I may know so ask me.