I'm not completely convinced, but you certainly raise some good points.
However, I do have some questions. What, exactly, is it that you say "maintains its behavior" in that way? Because from my perspective, what we have is not obviously a single thing that can be said to track across all those shifts for a given individual or family, but rather a metric that maintains a high degree of stability from generation A to generation A+1—and thus, a moderate degree of stability to generation A+2, and so on.
Furthermore, my understanding of that 0.7 number is that it's true for the US, now, not something that's static worldwide and across history.
Finally, I would note that there's probably just as much publication in conclusively proving it is genetic.
I don’t know exactly what the contributors to high social class are but intelligence and conscientiousness are no doubt among them and they’re highly heritable.
There are aspects of being middle class that come for free, they’re the ones children are born with. That’s why adoptive siblings are as similar in their educational outcomes as strangers and biological siblings are far more similar.
The mechanism is genetic, not cultural. If you can disprove it there’s a lot of publications in it for you.