The problem with government funded health care is what you can't see. True market based competition leads to lower prices and greater innovation in all observable areas over time. That would mean more access for more people in the case of health care, but instead with deep government intrusion for decades, we have seen ever increasing prices and sluggish innovation.
Libertarian philosophy has a lot to do with a belief in sound market principles. Why should all that go out the window for health care?
The problems facing health care today, in my view, are largely a result of the free market being trampled on. People who don't really understand market principles will tell you the exact opposite, that health care has been nothing but a free for all, totally unregulated. Do you believe that?
It sounds like you might generally agree with libertarian ideas and some research on scholars like F.A. Hayek and Ludwig von Mises might help focus your libertarian perspective.