Different jurisdictions are handling that different ways. I'm not sure how NY is handling, my only direct experience is how Delaware is handling it.
In Delaware there are exemptions to the ban for Fraternal organizations such as the Moose and the Elks and the Lions, Volunteer Fire Companies, and VFWs and American Legions.
Private clubs are practically non-existent in Delaware, but the few that are (mostly country clubs) are exempt.
Opening a private club that serves alcohol is practically impossible in Delaware. If for example Joe's Bar & Grill wanted to become a private club they would have to show prove that the organization has specific membership guidleines, is not open to the public, has a specific goal, and can produce a minimum of 6 months worth of meeting minutes in addition to their by-laws. And while doing all of that - they would be unable to have a liquor license, have to pay a $500 non-refundable application fee, and go through all kinds of other hoops and loops.
In other words, the types of private club you are proposing is pretty much out of the question in Delaware.