I generally agree with the point behind Mankiw's assertion that a public option destroys competition if it's publicly funded, and is no better than a nonprofit if it's not publicly funded. There is one point, however, that's being missed with that point.
A plan that covers pre-existing conditions, but otherwise is not subsidized in the slightest, will be competitive for people without pre-existing conditions who otherwise receive no coverage for it, but also serves as a safety net for people with these conditions.
I'm not taking a strong stance here, I tend to think that this type of option is good because it helps people deal with medical conditions that they aren't in a position to fix (presumably) but I could be convinced either way.
The much bigger problem with the public plan proposed is that it looks like it will be subsidized in other ways as well, and, even worse, it doesn't charge different premiums for people based on behavior and lifestyle. It also will not cover promising upcoming technologies because they aren't yet cost-effective, stunting their development (see my post immediately prior). Basically... there are major problems with public healthcare. It's just theoretically possible to have a competitive but still publicly-funded plan.