A friend points out that housing developments are often named after what they destroyed to put it there. "Shady Creek", "Deer Run", "Pleasantview"...

Name for the business you want rather than the customer base you have.

edit: apologies for the jokey answer but it's a serious point. I don't think it's an ironic name for a private company at all. If I'm selling healthcare services of course I want 100% of people buying them from me, ideally with the only competition being my subsidiaries.

They can always hope for the sweetheart public private partnership if we ever get universal health care

How so? "Universal healthcare" just mean everybody has healthcare. It doesn't mean it has to be provided by, or be paid for by the government. Several European countries (eg. Germany, Switzerland, Netherlands) all have universal healthcare that's at least somewhat privately funded.