The third is not a majority. The first link says 57% were not born wealthy.
>>> And when I look around at the rich people I'm familiar with
Anecdote is not data. I don't know where you live, but depending on it and your social circle, the minuscule number (compared to the whole nation) of people you can observe personally would be very different.
>>> Often it was a right-place, right-time thing.
It was. So what? Nobody claims billionaires are superhuman. The claim is only they deliver (or delivered) value. Did they come into position of being able to deliver the value by luck? Maybe. So what? If you per chance found a deposit of gold or discovered very valuable technology - they fact chance played role in it does not change the price of gold or the usability of technology.
>>> If Bill Gates had been born 50 years earlier, for example, it's hard to imagine him becoming a zillionaire.
Sure, so that guy everybody mentions would be called "John Walles" instead, it's no different.