In a democracy the lowest common denominator of citizens determines policy. They are easily manipulated by moneyed interests and always insanely uninformed (not because they are stupid but because it is impossible to know everything). Knee-jerk, short-term policy is a given.
Private insurance companies are held accountable by the customer's ability to switch plans.
I understand that billing is an incredibly frustrating issue, but forcing people to participate in one master healthcare system would be devastating to people battling poorly understood diseases. People in this situation desperately need choice in their healthcare options.