That's important, true. But that's not the only thing that would increase the amount of healthcare available to people. Dean Baker, for example, makes the important point that we could greatly increase the supply of doctors simply by allowing qualified foreign doctors to be certified in the U.S.[1] Lower class workers in industries like manufacturing have had to compete with foreign workers because of free trade, but upper class workers benefit from extreme protectionism that shuts out foreign competition. Protectionism is also why drug prices are so high; if people were allowed to import drugs from places like Canada, the cost would be substantially cheaper.
[1] http://cepr.net/blogs/beat-the-press/more-of-free-trade-in-d...