Your data shows that Americans buy more than one gun generally...including non-efficient weaponry, and allows multiple counting. Look at Cheeks...he has 20, but not one of them is an "efficient weapon" which is what we were talking about. I inherited a bunch of "cowboy guns" (1800's type things) which would drive up your data in a way that is also irrelevant to our discussion. Your data is misleading, and it does not mean that "efficient weaponry" is more readily available in the US to such an extent to explain the greater use of such weapons over the Swiss.
The Swiss have access to AR equivalents as a matter of right (essentially..."shall issue" purchase permits...and a decade ago even that wasn't required), but they even have ready access to fully automatic weapons (super-efficient weapons, to use a variant of the type of language you are using). But also, the Swiss data is unreliable. It wasn't too long ago that all able bodied men were required to have automatic weapons in their homes, ready for use in case of invasion. (The Swiss have a culture which thinks about such things.) And, they were apparently not tracked in public data. In 2008, the Swiss started tracking firearms, and they only tracked arms which changed hands and were voluntarily registered by the owners. The data there is suspect.
Either way, they have as easy access now as we do. That tells us that something else is in play. Perhaps they treat mental illness differently. Perhaps they have a homogenous society that isn't dividing itself on racial lines for political benefit. I don't know. But "efficient weapon" access is at least identical, and even leans toward more access in Switzerland than the US.