America has had multiple attempts at solutions for healthcare over the years, each started with good intent and then waylaid by various causes to produce what we have right now.
A sibling comment mentions political compromise to pass the ACA, as an example of this.
Another example is that HMOs were started with inherent goodness, but got “corrupted” (in my mind) by profit seeking.
To directly answer your question: a core tenet of the Republican tent is minimal government involvement in day to day lives of the citizenry. Ergo, the Swiss system won’t work because it involves a lot of bureaucracy. Republicans link bureaucracy to cost, and feel this is not an appropriate use of tax payers dollars.
The holes in this political doctrine are not part of my answer here fwiw. Please no “but…” comments to that end :)