About 20% of millionaires inherited their wealth. [0] (It may actually be far less than that, if you interpret "inherited their wealth" to mean "inherited over a million dollars." [1]) Almost 70% of those with ultra-high net worths are self-made. [2]
These articles are all exceptionally light on details.
A good example: Zuckerberg is considered "self-made," right? Except he had a wealthy father (dentist) with wealthy social connections who funded his company. It is at best misleading to claim Zuckerberg is "self-made", in that it completely elides substantial luck and money he did not, in fact, earn, to get where he is.
Zuckerberg certainly has the benefit of having parents a few standard deviations above the mean who could afford to send him to a nice school a few standard deviations above the mean and coming of age at exactly the right time that the internet was about to become ubiquitous.
However, compared to others, he is pretty self made in the sense that he did the grunt work to create his website and get it off the ground. Obviously, almost no one vaults themselves to the top with zero help, but Zuckerberg is a far cry from someone who inherited a trust fund and then placed various bets hoping to hit it big by funding someone else's work.
"Self-made" is one of those terms that one should probably never use, since it rankles otherwise generous readers. If you can get past that indiscretion, I think you'll see that I didn't actually make any claims that contradict your rebuttal.
A good example: Zuckerberg is considered "self-made," right? Except he had a wealthy father (dentist) with wealthy social connections who funded his company. It is at best misleading to claim Zuckerberg is "self-made", in that it completely elides substantial luck and money he did not, in fact, earn, to get where he is.