I'm not arguing the US system is better or even close to as good as the Swiss system. We have a lot of problems, and overpay flagrantly. I'm saying the Swiss have a system that closely resembles ours (unsurprisingly, since ours is patterned on theirs) and theirs appears to work. That tells me our problems are orthogonal to the issue of private versus public insurance.
I think the weak penalties for not carrying insurance are a serious problem with our system, but you are going to have a difficult argument to make if you want to attribute the hundreds of billions of dollars we overspend in health care to just that factor, or even significantly to that factor.