This is one of those things that is worse in Europe than in the US. Inequality is perhaps less in absolute value, but it's more sticky. The upper class of the US has plenty of immigrants and people who were born poor. Yes, being born rich still provides advantage, but it's clearly not impossible.
In the EU, the upper class has
zero immigrants
very few people born poor (upper middle class minimum) (out of the top10 riches Frenchmen, one was born to a family with less than 100 million francs at the time they were born. One. And he wasn't poor by a reasonable definition at all)
Of all OECD countries, only in Slovenia, Chile, Italy and the UK are the intergenerational correlations between the earnings of fathers and sons stronger, for example.
The rest of the OECD, including most of Europe, does far better in that respect.
Representation of immigrants is something I haven't looked at, and it may very well be that the US does well there, but there are a number of confounding factors, and it in any case doesn't change the fact that social mobility is poor in the US.
[1] http://www.epi.org/publication/usa-lags-peer-countries-mobil...
My uncle became a surgeon, and my parents become engineers, through the means of a free public university, which propelled them to upper middle class.
There is still much to achieve in French social mobility, but these kind of stories are plenty. Having spent some time in the US now, and seeing the stories of people who go on to go to engineering/medical/etc schools, I can't say that I see comparable things happening. Everyone I know who went to med school comes from upper middle class family AND had to take on tons of debt.
You shouldn't be looking at the top 0.001%, but the top 10%, perhaps even 20-30%, and how they live/what kind of social protections they benefit from.
The reality is that if you were born somewhere poor, odds are overwhelmingly that you would earn a tiny fraction of what someone born somewhere where they are provided proper education and an environment where better alternatives are available.
So maybe you are compensated in proportion to the value you generate, but the problem with this is that this is only superficially connected to the effort you put in and how effective you are.
Some people who do well do so in part because they put in exceptional effort, but the biggest predictor of your earnings and success in life is how where you live and how wealthy your parents are.
This is the problem we have with inequality: It has very little to do with effort or skills, and a lot to do with pure chance.