However, I say that as an aside. Jason's question seems to me to be more about golden age greens. I suspect that the answer is yes they do on occasion settle and indeed droop. I was a member at one Old Tom/James Braid course where a greenside bunker was filled in because the green was "sliding" into it.
The soils would have to be pretty awful for that to happen.
I can think of one similar case in my work: the 3rd green at Stonewall. We pushed some of it out into what was a natural swale, and we didn't have a lot of extra material to hold it up, and it had a lot of slope in that section [3% or 4% from memory]. Over time it has gotten more severe. But I don't think it's actually settled . . . I think that topdressing has built up the green a few inches over twenty years, so that the edge going down into the swale is that much steeper.
It is way way more common for greens get built up from topdressing over the years, than to settle. Most of the greens at Crystal Downs are at least 6-8 inches higher in relation to their surrounds than when I joined the club, as they try to build up sand on what was very poor original greens mix. Where there used to be a flat collar, there is now a steeply pitched one, or they've moved the green edge a couple of feet uphill to compensate.