Phil,
I think you hit on a bunch of it. He has a small portfolio of courses, so the number of courses he has on most of the lists is a significant percentage of his output. Flynn only had 60 original designs with 8 never constructed. He was involved in 21 redesigns of varying degrees. Of this work, 13 courses no longer exist.
The Depression halted most work. Flynn began work on only 5 courses after 1931. He was not a self-promoter. I've only seen two advertisement that mention his golf architecture business. These ads were actually promoting the basket golf standard that he patented in 1916 and not golf design.
Two of his construction employees, there were no other designers but he alone, claimed design credit for two of his best courses, Shinnecock Hills and Indian Creek. Most of Flynn's courses were very private and nearly all are geographically concentrated in the mid-Atlantic region. Most of his business records were tossed out by his wife upon his death. Thankfully she gave the drawings to Wm Gordon and he faithfully kept them stored for us to use today.
His design style, as you suggest, is overwhelmingly natural looking that it is hard to determine what, if anything, he did on a site. A budding "architect" mistakenly thought most of the Cascades was natural instead of it being one of the most manufactured sites ever developed for golf.