It is absolutely possible to get to university after only finishing Hauptschule. You just need to go to BOS after finishing your apprenticeship and then you can achieve a fachgebundene Hochschulreife (maybe even allgemeine, im not sure) and attend University. Few people do it, because the desire is not there, or maybe not the tenacity to study further after already having trained for a job. Also you get Kindergeld and Bafög while studying.

And when one considers that a child’s educational performance is directly related to both familial socioeconomic status and parental educational attainment,

This is true and criticized by PISA every time.
I think it has a lot to do with how much the parents value education. east asian immigrants are famous for how much emphasis they place on education and as a result get into university. The only thing that would help immediately (i.e. does not require behavioral change for a large portion of the population) would be to separate kids more from their families via Ganztagsschulen, to weaken this influence.