I know libertarians generally support completely privatized healthcare. And that in a truly free market, everyone would be able to afford their own plan, choose from different options that suit their needs, and no one would be missed out on due to the fact that an insurance business would identify the need and fill it.

This is a very common misconception of the libertarian model.

When you come from the left/right paradigm you tend to believe in absolutes, and tend to believe in ultimate solutions. Libertarians, and Libertarianism typically doesn't believe that.

Libertarianism offers a more free and just solution, but not an ultimate or absolute solution.

With health care. It absolutely IS a scarce commodity. There are only so many doctors, only so many pills, and only so many beds. etc.

No matter what we do, or what political system we have in place, some people will get care and some wont. No amount of money will create new resources, no amount of money will create new doctors, and no amount of money will keep people alive longer.

With heath care as a scarce resource, how do we determine who gets the care and who doesn't.

If we let the market decide, then people who can afford it get it.

If we let the "people" decide, then we are basically electing a small group of people to pick who gets the care

So how does a society justly determine who gets scarce items/treatments?