On the one hand, you do really want people to be relatively insensitive to the cost of medical care they actually need. When you need it, you get it. On the other hand, that price insensitivity, applied indiscriminately, leads to the massive ballooning costs we've all seen. I don't think there's a way to have full price insensitivity and to keep prices in check - which means that we either just spend whatever our doctors ask (I'm not interested in that), or we start making people more price-sensitive to allow those downward pricing pressures to be exerted.
Fundamentally, you don't get away with increased price sensitivity without someone yelling about death panels. Prices are how we allocate scarce resources, which means they determine who doesn't get things. We can either pretend prices don't exist (and get the status quo), or we can acknowledge that they do and that sometimes, people can't or won't pay those prices. Neither is emotionally attractive, but I think being realistic about the existence of prices and allowing consumers to actually exert some market pressures is preferable than just being held hostage to whatever the insurers demand we pay this year, just so that we don't have to think about costs at the point of service.
They are solvable, but with regulation, and they will sometimes get an edge case here or there wrong, which will inevitably result in a low-effort, sensationalist news article which will decry the situation, degrade people's support of the systems, and damage the institution.
Maybe I've misunderstood you but how can there be no way to achieve what almost every other country has achieved?
In Singapore, for example, Medisave only covers drugs that the government has deemed cost-effective. Need a high-cost experimental medication? Not permitted. Denmark has notably bad cancer survival rates, much a consequence of long wait times for screenings and treatments. The US, by comparison, has some of the highest cancer survival rates in the developed world, arguably because Americans can (and do) spend on tests and screenings at far higher rates than other countries. IIRC, Americans get medical imaging scans at a rate roughly 100-1000% higher than patients in other first-world nations. Most of those scans don't find anything, and if you're a government bean counter deciding how to allocate this year's medical budget, they're going to be high on the list to cut or ration - but when they find something, those early warnings and immediate access to aggressive (and expensive) treatment make a significant difference.
I bet there's a lot of people who would take this. And it would both save money and make a pretty big difference in the lives of people who still have a lot of life left to live.