Yes, you wonder if so many people would bother to incur the huge costs, if they realised the very limited value of many of the degrees earned. I realise many people would like to have the university experience, but it is a very expensive experience if it doesn't really enhance your job prospects.
I can appreciate what the study is showing. In order to get certain grades, children from state schools generally have to work harder. Perhaps it is due to the larger classes, less spoon feeding teaching methods, and less focus on the university application process. This means those who achieve those grades have 'more about them' than their private school counterparts who achieve at the same level, but have had more help to get to that level. Perhaps the natural ability therefore of the state school educated BBC candidate is higher than that of the private school BBC candidate. Interesting that there is less of a difference at the higher grades. Seems to me, that the standard needed for an A grade these days is not actually that high.
And perhaps more determination is needed from many state school pupils to achieve at an equivalent level. Those who have that determination are also able to apply it at university. Perhaps if higher education is the norm (for private school pupils) and success in life has come easily, they might show less dertermination and make less effort, because they are not so used to it. Sweeping generalisations obviously, but might be broadly true.
The thing is, the private school pupils are still advantaged at the end of it all. Their private school experience got them higher grades than they might have achieved elsewhere, so they got into the better universities. They also had the good university advice which directed them to the most esteemed universities and most highly valued courses. The fact they were at the right schools, universities and on the right courses all looks good on their CVs, even if they have not done quite so well at Uni.
More state school pupils, with similar or even higher grades may well have gone onto do less well regarded courses at less well regarded universities. The fact they got a higher degree classification at the end of it might not count for much, because of where its from and what it's in. We know that a number of state schools have given poor advice about the impact of educational choices, ranging from not ensuring bright children choose facilitating subjects at A level (which then means they don't get the offers at the top Unis) to not making students aware of the pecking order of courses and universities. These disadvantages are greater in poor performing schools without a history of sending pupils to the top tier universities. Where parents are poorly educated or unaware of the system, they are unable to help beat these issues. Admittedly, there are state schools that offer good advice, but it is not universal.
Sorry so lomg!