It's simple: anything over $X is covered by the state. Period. States that spend more than some amount set by the federal government gets eligible for federal assistance. States that have an abnormally high percentage on a regular basis gets audited and those audits determine if they get assistance from Sam. This would cover Alaska where all emergencies tend to need an emergency helicopter while states with high substance abuse problems like West Virginia are left to foot the bill with a sour taste for not implementing better social programs to address those issues. Audits would also force medical facilities to curb inflation and discretionary pricing (such as charging insane amounts of money like they do when insurance covers the bill).
Not insurance and not completely funded by taxpayers. If you have a life destroying health expense, the community is there to make sure your life isn't destroyed and that you have to choose between little Jimmy's college education or your life. Everything else is your responsibility and so incentivizes healthier choices.
It's a step in the right direction, but obviously lots of effort still needs to be made to remove distortionary policies that reduce the efficiency of healthcare. You're right that the reliance on insurance for routine care should be reduced. A good analogy might be to having automobile insurance cover oil changes: there is no reason for covering routine care since this can by definition be planned for and saved for accordingly. Putting transparent prices on these services (which would no longer be tied to your employer) would be great for citizens to choose between physicians.
Some level of coverage for catastrophic illness/injury seems appropriate though.