Earlier quoted context omitted.
This is not your main point, but in the spirit of Americans overestimating social mobility, you may be overestimating "family values" among the working class. According to the 2015 US census, the share of people who have been divorced at least once is 46% (poor), 41% (working class) and 30% (middle and upper class). Similarly, people who report they are "very happy" in marriage are 54% (poor), 56% (working class) and…
I think his definition of working class is perhaps different to the definition in your study. It's one of those terms that are loosely defined (at least in certain societies) and mean very different things to different people. Is an electrician earning US$80k/year middle class because he's well compensated for his work, or working class because he's doing manual labour?
In the 90s, I remember "family values" having been used by politicians as a polite euphemism for the very narrow "straight, Christian, atomic" view of what a family should be.