Turning the argument on its head a bit, my son's school is not disruptive at all without the level of discipline/ authoritarianism (particularly outside the classroom) that Michaela operates with. Why is that? Is it down to cohort, fee paying or cultural capital? Are there schools in London with different approaches to Michaela with and much perceived success?
Children's behaviour can have many factors at its root:
Poor nourishment - most richer children have fairly good diets. That is not always the case with those in very poor households, where parents may not have the time or knowledge to make healthy meals. Michaela has banned them bringing in any food or drink. This cuts out all the cans of red bull and bags of sugary cookies that constitute the diet of some children. A lack of healthy food can have an impact on behaviour.
Poor role-modelling
People who have been successful at school and have made the most of their education tend to be more positive about education. There is a big swathe of people who think it's a total waste of time and would rather their kids were working. Among MC families there is a tendency to have more of a "delayed gratification" - and so the importance of going to school so it builds up qualifications which lead to uni etc is ingrained. Poorer families need the money sooner rather than later in order to keep going. Kids soon feel that the work is "irrelevant" and "pointless". They are less likely to appreciate the importance of a rounded education ("why are we learning about religion? I don't want to be a priest"). At Michaela they apparently are constantly talking about their future at uni, something that they were not all likely to be invested in. With feeling their is a point, their behaviour will improve.
Supervision - parents in poorer families are likely to be working longer or more anti-social hours in order to pay bills. They don't have the same access to paid after school clubs or babysitters etc. By lower secondary level, some children (certainly on an estate near me) are out and about after dark with only pissed 15 year olds as their role models. This "street life" approach is a big problem because the norms of unsupervised teens don't tend to align with those of society as a whole. When teens have a good amount of time with socialised adults, they learn the different "codes" of behaviour. For example, your son would be likely to know the different code needed when he talks with his friends as when he talks to teachers. That's not always the case when a teen grows up with only really the "code" of the street.
Michaela are trying to combat this with the whole making them shake hands and look people in the eye.
"Authority" and "doing what you're told" are pretty uncool in society today. TV shows mock those who are compliant and obedient. Punishment is barely mentioned. In many schools, a response to asking a student to do some work might be: "errrrr no mate". "Don't think so 'Rebecca'!!" etc, to screams of laughter from others. Some parents would also support this "just a bit of banter"
You mentioned cultural capital - yes this can also be a factor in that teachers make constant references to this. Without any books at home or any visits to any theatres or galleries etc, any allusions to this will be lost on those who haven't been part of this, leading to further exclusion from their education.
In fact, the group of students performing far worse than any others is white working class boys.
Fee-paying - if parents are paying, they are far more likely to be very invested (too invested sometimes). That's absolutely not to say those parents with kids at state schools are not invested, but it's less likely to be the case . They therefore tend to want to feel that they have made a good choice with their choice of school.