At least for children who grow up with married parents, there's still quite a bit of income mobility.

Such children born into the bottom quintile of income, are pretty likely to end up in any quintile. The probability of ending up in each quintile from bottom to top, is 17%, 23%, 20%, 20%, 19%. Seems pretty fair to me; totally fair would be 20% chance of each.

It's a little skewed for people born into the top quintile, but not so much: 10%, 13%, 21%, 27%, 30%.

Source: https://www.brookings.edu/blog/social-mobility-memos/2014/08...

This sentiment is the driving force behind the popularity of many crypto currencies. Crypto enables class mobility where there was none before. For instance, purchasing $1,000 of Nano (XRB) at 10 cents (4 months ago) would be around $87,000 now if it didn't get stuck on bitgrail.

You can trade penny stocks in the normal market, too.

That's demonstrably false. You could make an argument that on average it has become harder to change your social class, but plenty of people start life poor or middle class and end life rich.

It's true to a first order approximation.

If there were no exceptions, then how could those born into wealth persuade themselves of their own moral superiority?