I clicked through the link and looked at what data was available--looked like a lot of broken links. But, depending on your definition of mountaineering, as someone with a fair bit of experience in the area, I simply don't believe that mountaineering at any significant level is safer than driving whatever sort of normalization you settle on. The latest fatality stats for US roads is something like one per 100 million miles.
However, as both you and the link suggest, it's essentially impossible to normalize. I don't know how I'd even draw a hard line between a "walk in the woods" and "mountaineering." But do whatever works for you.
I think "mountaineering" is probably more dangerous, especially once you get into the alpine.
I do (and love) outdoor climbing and even a bit of mountaineering, but it seems like most accomplished climbers die in the alpine. Some of what people do makes Honnold's famous free solo of El Cap look tame by comparison
It's essentially impossible to properly normalise across hours of doing the activity.
Those stats, are at best, an estimate but it's exceptionally favourable to mountaineering.