How are the working classes supposed to do that? From what?
There isn't a general working class that can't prioritise and provide extra educational support for their kids. In the past, workers' libraries, grammars and night school helped achieve some social mobility and more recently it's been a clearer part of some immigrant cultures to educate their DC to succeed beyond what was possible for the parents. It's not a matter of money, but a matter of aspirations and priorities. Encouraging, motivating, making time for extra study at libraries or online, which are accessible for all, if you can get past the hurdle of being intimidated or thinking it's not for you and the school should do it.
By saying this, I guess the chief is highlighting the known issue that it's not just down to schools but that parental involvement and support for the DCs education is a vital factor in social mobility. Which doesn't mean paying for tutoring etc, but on a fundamental level, not assuming the school will do it all and that's the DC education sorted. The tone of the OP suggests the very opposite - that it's entirely the state's job - a misconception that needs dispelling if social mobility is to gain any traction. Obviously the state has to do its bit, but parents must too. Sadly there will still be parents who are unable or unwilling, but that's the smaller subset and not the entire working classes by any stretch. The more everyone else can step up, the more state resources can target those in the most need of support.