The general public has access to far greater growth investments now than they did prior to Social Security. It is possible and feasible to take a small fraction of your income, invest it in a well diversified portfolio, and have a lump sum enough to guarantee a stong income for life. A 3% contribution into a diversified stock/income portfolio in a Money Purchase Pension grew to a lump sum more than adequate to replace my father's work income.
The Enron people who did not diversify their retirement accounts and held large percentages of a single security were stupid and irresponsible. While it does not excuse the executive fraud, those people had the power to mitigate the damage.
The other advantage to private accounts is you have the ownership of the assets to bequeath to your heirs. Under the current Social Security system, you die at 57, and you don't get squat for the 30-40 years you've paid in. My father dies at 57, and my sister and I were able to receive the proceeds of the Money Purchase Pension to jump start our financial security.