I basically agree with you that we overfocus on the single-payer vs. private distinction. But I think there's more to it than that: when you get down to it, most Americans who want to reform the US health care system don't
really care whether it's public or private. They care whether it's
universal or not. The Swiss system is universal health care because the basic plan is capped at 8% of income at the federal level, with cash subsidies given to those with incomes lower than that.
By contrast, Medicaid in the United States is administered to the states, who have broad leeway in determining who is and who is not eligible for the system, especially after National Federation of Independent Business v. Sebelius. Income level alone does not necessarily qualify one for Medicaid. Medicaid is not considered universal health care, even with the ACA, and that makes a huge difference.
It's not so much that Americans don't want the Swiss system as that we're missing the crucial part of the Swiss system that makes people feel secure that they'll be insured.