This is a great idea! In theory it probable would work. The only problem I see with a government ran self funded retirement system to replace and bailout the social security system is this.
Currently, NO elected official has the intestinal fortitude to propose a solution such as this and gain enough backing from both parties to get it passed. Even if they did, it would be regulated or taxed to death or worse at some point they would not be able to keep their hands off the money and spend 250 million individuals retirement accounts as was done with the social security trust fund. Your idea is a realistic solution if implemented and left alone it could work.