What about people born into wealth, who inherited the money their parents or grandparents earned by doing something useful for society? Where do they fit into your picture?
On the flip side, what about people who cannot find any job that pays enough for them to save money, who are literally living paycheck to paycheck. Do you want to live in a society where we have a class of people who have no social mobility at all, and who lack the power needed to change anything about that?
Ayn Rand's image of the heroic capitalist who amasses a fortune by solving society's problems is very unrealistic. Even people like Mike Bloomberg and Bill Gates, billionaires whose fortunes were amassed as a result of their own work, received help early on. Who do you think paid for Bill Gates or Mark Zuckerberg to go to Harvard, where they met the people that worked with them to build their businesses?