But Matt, how many normal club golfers, not private club members or the like, would have heard of Shinnecock and how many would have heard of Pebble Beach? Answer that question, and be truthful, and you'll have the answer as to why we say Pebble is an Icon and Shinnecock is not.
Pebble Beach has held 4 US Opens (not counting this year)
Champions:
Jack Nicklaus
Tom Watson
Tom Kite (certain outlier)
Tiger Woods
Shinnecock Hills has held 3 in recent times
Champions:
Raymond Floyd
Corey Pavin
Retief Goosen
Shinnecock does not have nearly the pedigree of champions that Pebble Beach does, and that is part of why it is so recognized. Also, it is seen every year in the Pro-Am. Shinnecock is seen once every 9 or 10 years by the general public and then it fades back into relative obscurity, likely just how the members want it. Certainly Shinny has more history overall, but how many average golfers could tell you the first 5 clubs that founded the USGA?
Oh, and one more little demonstration of why Pebble is an Icon: