I think those studies are fundamentally flawed.
If they had been done in 1950-1970 I believe it would have shown social mobility.
However, today, because there are so few grammar schools, it tends to be middle class parents who get their kids tutoring and push them to get into the grammar schools, so they get the places.
As a student at a grammar school, I can safely say that at least 90% of the students here are middle class - aka there is very little social mobility. If grammar schools were a lot more common and kids were given tests that couldn't be trained for then many more working class kids would go to grammar schools, massively increasing their usefulness as a tool in social mobility.
Chat about results here