Public chat about social inequality and social mobility is typically bollocks, 2 reasons being ignorance about what happened in the past and a failure to acknowledge how classes actively strive to reproduce themselves as best they can.
Starting with the past, if I remember my (very simplistic) reading of John Goldthorpe correctly, essentially post war Britain saw the number of middle class jobs grow faster than the middle class could breed to fill them; lo and behold - more working class people got middle class jobs. Now? Less so, especially now that so many middle class women have actual careers.
Then there’s class reproduction, which for someone like me is about giving my child every practical advantage I can. This includes stuff like having the cash to move into the catchment area of a good school, paying for additional tutoring, a private education etc. Then there’s the practical stuff like what to wear for a job interview, how to write a CV or what careers to consider (and which uncle or aunt to talk to about them beforehand) etc., However, its the "soft" cultural stuff I remember most from my childhood, like exposure to ways of discussing and arguing and, the most powerful for me, the basic assumptions and expectations about what I would go on to do with my life and where.
Oh and one final thing, the chat politicians come out with about training and skills and what no ignore the reality which is that there will always be low skill, low wage work. Plus, the training and skills chat buys into the high skills get high pay rhetoric used to justify inequality.
What I take from all of the above is simple. The first is you can’t change the structure of occupations overnight. The second is that greater social mobility would entail taking on middle and upper class parents who are all in favour of mobility when it involves their kids getting on, but less so when its a threat from "below". So debate all you want about education and what no, but in the meantime just introduce a living wage fer chrissakes.