Several of the holes at ANGC have short game elements that remind me of things I've seen at Pinehurst #2, Dornoch, and as you already mentioned, St. Andrews. The short-game options around 1, 3, 5, 6, 8, 11, 14, and 17 all bear resemblance to things you'd see on those courses. Obviously the weather conditions and the American style of golf do not allow ANGC to be played like Dornoch or TOC, but if I'm trying to compare different aspects of courses, those are a few that pop up first in my mind.
Though I've never been there, and thus don't know enough to compare individual design features, you'd have to toss Cypress Point into the discussion as well. If I recall correctly, Bobby Jones first discussed ANGC with Mackenzie at Cypress while it's construction was in progress.
My last thought though is that you're dead on with The Old Course as Mackenzie and Jones both had a particularly strong affinity for The Old Course and the fact that much of its beauty was because it just looked as if it had always been there.