The nominal GDP in Switzerland is actually much higher than in the US, it's just that the country is way more expensive to live in. Therefore, when adjusted for cost-of-living (PPP), the GDP are about equal.
Of course, the absolute amount of money per capita is still much higher, which lowers the relative cost of, say, medical equipment.
The private healthcare in Switzerland is also compulsory and insurers are forced to offer a basic, non-profit plan. Furthermore, individual payments may be subsidized by the government. It's therefore not that different from Obamacare, nor are private insurers left to the devices of the "free market".
You point at some facts that happen to be similar (when adjusted) for the US and Switzerland, but it's rather arbitrary. What the hell does gun ownership have to do with healthcare? And of course, like most industrialized countries, there are "lots of cars"...
The absolute difference in wealth doesn't affect just medical equipment, it affects practically everything besides labor. One of the reasons why labor is so expensive is the high taxes (VAT etc). When the government subsidizes that expensive labor, it'll get a big part right back.
You also edited your post to agree more with mine (i.e. Obamacare), when the original point was more like "private insurance can work look at Switzerland!".
Yes, it can work, if you're wealthy Switzerland and you regulate the fuck out of everything. Touche...
I just like to use it to point out how the common American argument against universal healthcare don't really apply.
"it's socialist!" - no, you can have a system where everyone pays in according to insurer determined risk factors, pay less overall and still be within what people at minimum wages can afford. Below that you subsidize with taxes, but that costs very little compared to total health spending.