Sean:
A number of years ago my Dad, who was an avid golfer and avid traveler, went to London and surrounds with a group known as the Cenral Connecticut Golf Ass'n. The trip was arranged by a British fellow who had moved to the States. There was another ex-Londoner in their group. While they were traveling on the bus my father asked what the Central Connecticut Golf Assn was. The guy who arranged the trip told him that without that type of cover he and the other ex-Londoner would be let on the courses. They said at that time that they lacked station in Club circles.
I would think by now that that has eased to some degree.
Exclusivity at clubs in America is desirable. Most, if not all, the major clubs in the US do not need the money or the hassle. This makes for finding sources and contacts to get entry to these clubs. There are a number of clubs where you don't even ask.
It is pretty silly. The idea of exclusivity is passe. If you are determined, can talk the talk and walk the walk you will gain access pretty much everywhere.
I think that most Clubs would gain financially by setting aside times for visitors. In Pinehurst, some of the private clubs do a brisk business with non-members. They are referred by travel arrangers. This has greatly helped some of the clubs.