Peter
A few years prior to that article you quote, Low was railing against "Haskelitis" and encouraging existing quirky layouts to be remodeled and lengthened, presumedly to make them more "competitive." As a result we lost a lot of charming, challenging and interesting golf courses, at least in Britain. The ones overlaid in their place are mostly "better," but not as conforming to the ideal he seems to have adopted by 1917, well after the hoirse was out of the barn.
The only course I can think of that does (or even could) reach Low's ideal is the Sheep Ranch--partly because it is private and one can play it in solitude without bothering other players (as does the lone angler on the hidden stream), but also partly because it is not a "course" per se, but a collection of tees and green sites that can be played randomly. It is possible, as Tom notes, to create one's own private "Sheep Ranch" on most courses in Britain and Ireland, if you play very early in the morning or very late in the evening. I often do both on my holidays at Dornoch, in order to spend more time with my children in the middle of the day.
As to whether the dichotomoy (between competitive and non-competitive golf) is related to design/GCA, I would say no. Any course can be played contemplatively or competitively. One could have a contemplative round at Oakmont as easily as a competitive one at Mallow.
Finally, those (like Low and Behr) who think that any game of golf can be completely non-competitive ignore the fact that the essence of golf is trying to hit a ball at some sort of target--be it over a stream, at a green or to a hole. It is also about personal satisfaction and achievement. No golfer loves a foozle more than a clean strike, any more than an angler would prefer a cast which embeds a fish hook in his hat over one which caught the interest of a fine brown trout.....
Rich