The problem is that in the past year and a half the Federal government has stood behind the debts of not just Fannie and Freddie, but AIG, Bear Stearns, Merrill Lynch, Bank of America, Morgan Stanley, and Goldman Sachs--none of which bear any resemblance whatsoever to a "public plan." The government has stood behind Fannie and Freddie not because they were, before 1968, public enterprises but because they were--like AIG, Bear Stearns, Merrill Lynch, Bank of America, Morgan Stanley, and Goldman Sachs--too big to fail. The Treasury staff would have loved to have let Fannie and Freddie default on their bonds had they not feared the systemic consequences.
The fact that Mankiw can't find an example of his argument (2) makes me think that it is very weak, and that the real reason people oppose the public plan is (1).
I agree that the Mankiw's Fannie/Freddie example is emblematic of a weak argument against a public health insurance plan. But, I actually seriously doubt that Fannie/Freddie would have been allowed to fail their creditors under any circumstances.
Foreign central banks bought FF debt and MBS at least in part because FF securities were called agency securities. Subordinated debt on FF debt was not AAA, and so it is highly unlikely that foreign central banks would have invested in FF debt/MBS were it not for the ambiguous government guarantee, an ambiguity that arose at least in part because Fannie's debt was public before 1968 (Freddie was born in 1970). LBJ wanted Fannie debt off the government balance sheet so that the apparent Vietnam War deficit would be lower. Had FF been allowed to default on debt held by foreign central banks, Treasuries themselves might have been much less appealing. And if the government is going to make the Bank of China whole, it is also going to make grandma whole. It is striking that FF debt continued to trade at favorable prices even when they couldn't produce timely financial statements.
I have in this space argued that FF were not the primary causes of the crisis--they turned out to be slow followers of the "purely private" market in funding unsustainable mortgages. But let's not kid ourselves: the enterprises long have had an implicit subsidy, and they took advantage of it when they could.