It's whether your parents taught you to read by age 5, cooked vegetables for dinner, encouraged you to play an instrument, drove you to tournaments on weekends, picked a neighbourhood with an IB program even if that means no yard, paid for braces, etc etc.
Patents with low income can choose to do formative things, with personal sacrifice.
Anecdotally, social mobility happens when your parents choose to prioritize you, and you also somehow get introduced to the good life and determine to 'make it' at whatever cost.
(I'm not disagreeing. Intuitively I'd tend to agree with you that those things would have a measureable effect. I just don't trust intuition in this sort of thing. :) )
Malcolm Gladwell writes about this too, in Outliers [1] (chapters 3 and 4). His aim was to dispel the myth that success derives from intelligence.
[1]: https://www.amazon.ca/Outliers-Story-Success-Malcolm-Gladwel...
It's not the wealth. It's all of the cultural things that correlate strongly with wealth. Case in point: Two million Cuban-Americans came to the US with nothing but the clothes on their backs, within the last 60 years. Today, they've found their way to the middle and upper classes in about the same proportions as non-hispanic white people.