I think this is a disagreement about the definition of "mountain climbing." From the available statistics: 0.68 per 10,000 climbers in the Alps vs 1.5 per 10,000 US vehicles on the road.
High altitude mountaineering is considered an elite endeavor. Most mountaineering is not at high altitude.
That's fair. If you use any of the numbers closer to the activity described in the article (mountaineering at 22k') then you see the disparity. Even non elite mountaineering (mt ranier at 14k') has twice the mortality rate, according to this data.