Minimums are needed because cross subsidisation is rather integral to having affordable healthcare for everyone. Meaning those who are in the stage of their life that don't need much medical care pay more, but those that do are able to afford it.
I definitely agree with the suggestion that you need to break the link between healthcare and employment. Because there's obviously a very strong link between unemployment and needing medical care. I love living in a country with universal healthcare funded by taxation. Sure it means that I end up paying a lot more than I otherwise would, but it provides a safety net for everyone - including me should I happen to get unlucky and end up sick and unemployed. People don't end up having to declare bankruptcy because of medical bills or go without life saving medical care. And everyone is able to access a reasonable standard of healthcare (and the wealthier are not restricted from getting gold plated service if they wish to pay for it). But hey I realise that for a lot of Americans that concept is just communism so it'll never happen