Consumer choice and honest competition are indeed the foundation of a successful market system, but they are usually achieved without a public provider. We don’t need government-run grocery stores or government-run gas stations to ensure that Americans can buy food and fuel at reasonable prices.
One issue that Mankiw does not touch upon is adverse selection. The pool of the uninsured is quite heterogeneous, containing those who lack income, those who expect to be healthy (from all income ranges), and those who expect to be unhealthy (again from all income ranges). Insurance companies can collect limited information about each person who shows up as a potential purchaser and will try to limit their losses through denying coverage or by not treating pre-existing conditions. Knowing that the pool of customers will be skewed toward the unhealthy (and the risk averse), the insurers have to charge a higher price than they would if everyone was forced to buy coverage. This is a variation of the "lemons problem;" it also explains certain aspects of the market for used cars.
Two ways to deal with this issue without the government entering the health insurance business: (1) require everyone to have health insurance and (2) provide subsidies to those in certain income categories.