The advantages I had were basically four:
1. My parents taught me to read before I went to school. It is a cultural difference. Some people take it for granted that the schools will teach their children to read. Some people regard it as borderline child abuse if you don't teach your kids to read before they start preschool. My parents were pretty chill, but I could read before I entered preschool at age four -- they took that for granted as one of their parental responsibilities.
2. My father was an academic, I had early academic success, and I took it for granted that I was smarter than other kids. That probably had two causes. First, reading. I was taught to read early, and everyone in my family read. It was just something that one did. Second, I was probably exposed to a more analytical style of discourse at home than other kids were. So I was effectively "smarter" when it came to academic stuff, though stupider at other stuff.
3. I had middle-class expectations. Of course I was going to college and going to enter a smart-person profession. I couldn't understand why other kids didn't glom on to that possibility as soon as it was presented to them. Why did other, less enjoyable, more painful prospects seem more probable or more natural to them? I could see an easy path for them to lots of different things: a college education, a good job, maybe a professional degree. They weren't stupid, but they seemed to assume a "stupid person" fate for themselves. It baffled me. This difference is also cultural. It wasn't a physical or systemic barrier that somebody placed in front of them. It was a barrier that was psychologically engrained in them.
4. I understood how school and college worked. Actually, I didn't really. I didn't know that you could ask for extensions and so forth. I didn't know you could ask professors for help; I thought only obnoxious stupid bimbos did that. I could have known these things if I had asked my dad, but I didn't. But I did know that you were supposed to study and read and turn in all your work on time, and you were supposed to read beyond the assigned work if you wanted to do really well. And I knew you couldn't trust other people or do what was "normal" because a lot of people were going to fail and fall by the wayside. That cynicism, I'm convinced, would save a lot of people. A lot of people are good enough, smart enough, disciplined enough, etc. but at some point naively follow the example of someone they shouldn't. I knew better than that. You do the work, you read, you do better than everybody else because half of them are going to fail. That isn't intellectual knowledge or insider information: that is a cultural attitude, a different but equally important form of knowledge.
So yes, culture is a huge deal. My culture was "better" in the sense that it resulted in a more secure, freer existence for me.
I really don’t think that the ability of professors’ kids growing up to be lawyers or doctors really says all that much about overall “social mobility.”
It says social mobility can be achieved with no social connections and only a modest amount of money. It says social mobility is socially and economically within the grasp of a huge number of people who miss out for other reasons, and those other reasons are cultural.