Health care will always be a "problem" in any society where it is delivered through a third-party payment system. And it really doesn't matter if that third party is a government agency or an insurance company.
The basic dilemma is that the introduction of a third party into the process throws the supply/demand principles of economics completely out of kilter.
You have that right. The third party payments and arcane insurance codes, exclusions and other flim-flam have driven health costs of the charts. There is no market mechanism in place to rein in costs.