The use of the term links to mean a golf course goes back to the early Scottish courses. This type was built near the ocean on sandy land. The "link" below to Wikipedia gives some interesting history as well as the names of some famous Links. Note that two of them were in Southampton on Long Island. Could the Links Club in NYC be more of a generic golf-centric private club rather than relating to a specific country club golf course? I have no personal knowledge of the area that could help, I'm afraid.

I do remember that in the 1940's and 50's, the term "Links" was used for any type of golf course, even the Country Club in my little hometown in Missouri that had sand-greens and only 9 holes.