Your paper shows that some of them do. In all four tables (age/sex subgroups), the top quartile includes France, Belgium, Austria, and Switzerland, as well as South Korea. Germany is in the top quartile for females.
It's the ones that don't which disprove the hypothesis. Ireland and the UK both have pervasive public health systems, with nursing homes being a more common destination for the aged than living with children, but are only in the second quartile.