What you may be forgetting is the real, underlying reason that our society feels that healthcare is a responsibility of all of us.
We believe that no one should die of being poor.
Nor should they be incapacitated, unable to work or put through unnecessary suffering because of lack of money.
There are differences between our society and the one to our south. This is one of them.
We do not do the best job of managing the implementation of the basic principles but that is definitely where we start. This comes from the top down and the bottom up.
Where we might be going wrong is to allow this to become an idealogical purity issue. The tendancy is to think that everyone should have everything paid for by all of us. This is already not done or attempted but we forget that maybe we should think through what and who should be covered in what way.
Over and over you hear the politicians saying "No two tier system!" When we already have more than two tiers. Instead they need to work through what we really want to protect people from.