That's not entirely true but I believe it's more true today than it was 20 or 30 years ago.
For all their, significant, faults, when Grammar schools were prevalent they were a vehicle for practical social mobility i.e. people whose parents had done manual work getting well educated and finding professional jobs.
That's not true with Grammar schools today, their rarity causing entry pressure that has the effect of limiting the majority of students to children of the professional classes. Also, the Grammar system was/is hugely prejudicial. Although it was an avenue for social mobility that doesn't mean that it was an effective social good.
Increased university participation has increased educational levels but doesn't appear to have had a commensurate increase in professional labour either.
To some extent I agree with the article: it's not clear to me that social mobility should be an end to itself. However, I tend to agree with the New Labour idea of equality of opportunity in preference to the old Labour idea of straight equality. As such, the conclusion that changes in the ownership of capital will sort everything out doesn't strike me as having much evidence.
(And yeah, it's not completely true: I'm one of the ones who made it across. From what I can tell, there aren't that many of us about.)
Britain's meritocracy is indistinguishable from its old class system.