Yes, natural consequences is far too limited because in reality most natural consequences are far too far away to have any hope of even registering in a child/teenager's mind. Most of what we think of as good behaviour involves either empathy, taking another person's feelings/opinions/needs into account, which children can be very good at given the opportunity (but do need to develop these skills, which takes time!) or delayed gratification, the idea of denying yourself something in the short term for a longer term reward which is much more satisfying. The problem with this one is that children simply have not developed very far in this skill - and in fact teens tend to experience a temporary regression in it too, which is why parents often say that their children are lovely towards the end of primary school and then suddenly become challenging again as teens.
Going with the delayed gratification thing - we understand that things like tooth brushing, regular dentist visits and not eating too many sweet things all have a cumulative effect and will result in nice teeth which are easy to clean, fresh breath and not needing to have painful or frightening dental work done. This just simply not on a child's or teenager's radar. You can go on and on until you're blue in the face about cavities and bad breath but unless a tooth is going to cause them pain right now if not brushed, the vague idea of avoiding some dental work in the future just won't compete with the extremely minor victory of using those two minutes otherwise spent toothbrushing for something more fun, or the extremely attractive prospect of eating lots and lots of sweets.
I've developed my understanding of this a lot too because I was diagnosed with ADHD a couple of years ago, and delayed gratification is something which is very difficult for those of us with ADHD.
That said though, it's not a case of natural consequences being the only alternative to punishment and I think this is what a lot of people misunderstand. If you take away punishment, which is supposed to be a fix-it, a cure-all, you're left with a lot of different options which all work for completely different situations and different children and you have to combine them and make them work and be creative and it is hard work. It would be just as difficult to list all non-punitive techniques as it would to list all of the possibly ways children can misbehave.
I have issues when teaching too though - I am a language teacher. I am trained to teach adults who are mainly cooperative, but I've been asked to teach some kids' classes too, for which I have had absolutely no training except for my experience teaching adults and of course as a parent. And goodness, it is really difficult to manage some children's behaviour. Actually I'm somewhat forced to go no-punishment in this environment too because my employer ONLY cares about the reports of parents, so I can't be too harsh on them, I keep being told I'm allowed to send them out to calm down for 5 minutes but in reality I'm never ever backed up on this so it doesn't happen. I can't force them to do boring work because they don't do what I say anyway! So I sometimes end up shouting but mainly to make myself heard or get their attention.
I have to admit that I generally feel like my issue is that I'm not engaging the students enough, and if I could only make the lessons more dynamic and interesting, they wouldn't play up, I'm stuck in the same model of thinking that I shouldn't need to rely on punishment.
That said, if DS had been at school and we'd still been doing the non-punishment thing I wouldn't interfere with the school's methods. Absolutely not. I trust the school to use methods which work for them even if I might not agree. If I really strongly felt it was counter productive then I might explain why and see if they were open to trying something else but I wouldn't then simply hold my hands up and tell them he's their problem - I would have withdrawn him from the school if I thought their approach was making things worse.