The reason we have social security like it is is twofold: 1) current retirees are paid directly from our contributions. There is no pot of money to be invested and 2) we want to be able to guarantee a defined benefit that is not below a certain number. You can't do that with the market.
Between the pyramid scheme nature of social security, the high inflation potential of the USD, and the ability for politicians to leverage social security for short term approval boosts
You definitely can't do it with the government either...