Regulations in Europe also seem to have had the, I'm assuming completely unintentional, side effect of completely cementing the positions of the top of society in place. And this is nothing new, this actually predates modern democracy in Europe. It's been that way for centuries. In Europe the only time to leave, or join, the ultra-rich is during wars.
That the EU doesn't have unicorns is not an accident of whatever rules you like or dislike, it's entirely by design.
The problem with the ultra-rich is not that it's hard to get there, but rather that it is possible. Nobody is personally worth billions, period. The fact that some individuals get there shows a flaw in the system.
The second thing that I wanted to say is that even though there are examples of originally not ultra-rich people becoming ultra-rich in the US (e.g. Zuckerberg and Besos), the likelihood of this happening is almost zero. Why is it that people keep hoping that it may happen to them? We should not build a system for a handful to become ultra-rich, but for most to live as well as possible.
The underlying emotionally driven question is what did they do to deserve it. For all of the faults of Mark Zuckerberg or Jeff Bezos, I don't think I've ever heard of them being called lazy. Someone else getting ahead when they don't deserve it is universal. We get told life isn't fair and just accept it but I don't think I'm alone in never actually accepting it. So unfairness matters.
Amusingly enough, what the GP stated was true in 1910, but now the US is far more like pre-WW1 Europe in terms of distribution of wealth (c.f. Piketty's capital in the 21st century).
Social mobility index doesn’t really look at how easy it is to become very rich (I.e. get into the 1%). This is also explained in the methodology section of the article you linked.
That the EU doesn't have unicorns is not an accident of whatever rules you like or dislike, it's entirely by design.