What does a third world nation in which the vast majority of people live on subsistence farming have to do with the fact that the Swiss system intrigues me. I do, however, find it mildly terrifying that you are blaming death on lack of government.
Death is sometimes the result of lack of government. This is indisputable. Someone dies because they aren't treated. The government could have treated them.
Death is sometimes the result of government, too.
Also, the "government system saves people that otherwise wouldn't be saved by market system" misses the point that resources that would exist in a completely market system simply don't exist in a government system. Even with all the garbage systems of the world, this can easily be seen comparing, say, the high end of the US vs the high end of the UK, the German system vs the Swedish system, etc etc etc. Hell, in a lot of places (like Eastern Europe), you can pay a bunch of taxes for the privilege of NOT receiving treatment, so the "universal coverage" may as well not exist, and there is no private system growing via profits to cover everyone else.
