The problem with having the government do it is that they can be collectively less responsible. Think about it: instead of having a system you trust, you need to have a system that 51% of the voting public trusts. When that voting public consists disproportionately of old people, that means a system that pays out more than it can afford to, and redistributes income from young to old.

In other words, you get what we've got, which doesn't even work.