Thus, the choices, in his mind, are to get the public insurance, or get his own private insurance, and pay for the public insurance anyway. Obviously, if these are the options, nobody is going to pick the latter option, so we'll end up with only one choice for insurance anyway.
If these are the options, lots of people are going to choose the latter option. We can assume this because Americans are not that different to the rest of the world's population, contrary to popular opinion, and this is the way things work in many different countries.
The British NHS provides many things free-of-charge, and much else at greatly subsidised rates. Everyone has to pay for the upkeep of this through taxes. Nevertheless, a substantial proportion of the population (about 10%, I think) also pays for a private health insurance plan. They do so in order to get services which the NHS won't provide due to being too expensive or of dubious efficacy; in order to get faster access to popular services with long waiting lists on the NHS and/or because they believe a particular private provider will offer a higher standard of care.
Here in the Czech Republic, I can go to a government run poliklinika and pay just 30 crowns for my checkup (free until recently, health care reform here means increasing government revenue by bringing in new charges). Alternatively, I could pay much more at a private clinic. Despite the cheaper government option, private clinics have no problem attracting customers by offering a more pleasant environment, additional services and either better care or the impression of better care.