The issue is that income/wealth compounds over time. If people in the bottom 40% have very little to invest, they have nothing that compounds. They dont even have an opportunity to risk anything, or make a better decision.
The prices of the essentials are largely out of any individual's control. So any lack of ability to save / invest beyond spending on those is also out of their control. It's a vicious cycle, where excess wealth compounds, and income needed to exist dries up.
Also the rich often see a larger return to their investments than poor people, so having wealth seems to aide in your ability to make more money. Exactly the sort of thing that you implied was unjust.
And your other points about mobility I agree with and are addressed in notes on the site.