I can see how it all sounds like old fashioned discipline, Juule. All societies, even one in the Amazonian rainforests have a set of protocols, most undiscussed because it has 'always been like that' which everyone else around the child has already bought into so there is never a need for explanations, because like children in our society, their children instinctively prefer to conform.
When there are 22 kids basically following the instruction in a classroom, and one is pushing against the boundary, it is phenomenally easier to walk away and leave the child with the knowledge that he is to finish the exercise, or whatever. At home, it maybe brushing teeth. The child does not really need reminding of a sanction in the classroom in this scenario but allowed to be choose what they do next. What they tend to choose is compliance not to please me, but to not be different from the rest.
At home, I truly believe in not sweating the small stuff. There are few things which I think I would not accommodate according to the specific set of circumstances for that day. But it is also important to be mindful of what your long term intentions need to be. In a classroom environment, I don't have that luxury because children have a keen sense of fairness which is a double edged sword, so on a given day, I cannot break a rule for one but expect others to follow it.
When I walk away and begin to ignore, I am letting the child 'chew on' their choices: To comply or not to comply. School is not home, like you say, because there are sets of sanctions set by others, not the classroom teacher, but there are also subtler sanctions which can work before you have allowed things to 'go too far' and have to bring those sanctions into play.
Here is a way their fairmindedness plays into your hand too: I can't do that because that is not fair:As long as they are in the frame of mind to listen 'explanations' like, 'everyone else is expected to do x and you are asking me to treat you differently. to them/ to you/to me.' Note: kid has not asked WHY yet. You have provided a WHY in a general way, before you or he gets into a more recalcitrant mode.
When I notice unobtrusively that the child has complied, I don't praise or reward at this point, I just stop ignoring them and I treat them more inclusively. I don't praise what is expected of everyone. I praise, or rather, recognise when they have gone further then they have previously been able to go before. Not: wow! fantastic job! More like: that piece of work is really good because it shows more detail, insight etc or: I haven't seen you use that word/phrase before etc. At home the 'praise' may be: you tidied your room/ filled the dishwasher really quickly and so we could get outside sooner, blah blah. Praise, aka, the bribe is not the end in itself, it is a way of opening a two way dialogue which humanises me to the child and humanises the child to me.
The long term goal is to build a trusting relationship so the child is less likely feel like he needs or wants an explanation / a reward / do something with conditions attached in the first place.