Joe Nocera gets the story mostly right in his skepticism toward a bill put forward by Senators Mark Warner and Bob Corker that would replace Fannie and Freddie with a convoluted system of government guarantees. I would add three points.
First, the 30-year fixed rate mortgage is not necessarily dependent on a government guarantee. In the pre-bubble days banks did hold a substantial portion of the mortgages on their books. They also issued 30-year jumbo mortgages which could not be guaranteed by Fannie and Freddie, so apparently there is a market for these mortgages even without government guarantees.
Second, an important goal of policy in housing finance should be efficiency. In other words, the point is to make it possible for people to buy homes with as few moving parts as possible. This could probably be best accomplished with something like the old Fannie Mae.
In the pre-privatization period, Fannie was a publicly owned company that bought and held mortgages. There were no mortgage backed securities. There is little obvious purpose to move from a government company/agency holding mortgages to a government company/agency guaranteeing mortgage backed securities. In the latter case the government still has the credit risk, it only shields itself from timing risk (fluctuations in interest rates), but that is a risk the government has no problem bearing.
There are political reasons why the old Fannie system might be difficult to reinstate, but we can at least be clear that it would be the most efficient way to manage housing finance. Other methods are political compromises.
Which brings us to the third point, the waste created by these compromises is income to the financial industry: hence the argument for the sort of convoluted system of guarantees in the Warner-Corker bill.
So if we want the most efficient system that is politically feasible, it's probably best to just shut down Fannie and Freddie altogether. The 30-year mortgage will not disappear, although it may cost somewhat more.