In Switzerland (and also the Netherlands), the vast majority of this is covered by private health insurance, not via any sort of public health insurance. That's what sets Switzerland and the Netherlands apart from other peer developed countries, and they have results to show for it.
Regarding rare conditions, we agree — even the US has a public healthcare system alongside which covers rare conditions (e.g. end stage renal disease). US's Medicare Advantage system of private individual health insurance for senior citizens (with Original Medicare as a public option) is also a proven model with results to show for it. The CBO expects that by 2032, 61% of health insurance expenditure for senior citizens will be by the private sector (https://www.kff.org/medicare/issue-brief/medicare-advantage-...).
The intention of my comments was to make clear that in that regard, Switzerland is not an efficient free market private insurance paradise. There are lots of nuances to it, and the private insurance system in fact brings in inefficiencies, namely, a very complex and costly cost itemization and enormous amounts of paperwork and stress for doctors, customers, employers, and the insurances (for example, if there is an accident in a public place involving a car, there are at least one or two days worth of people's times just filling in forms)