The Versace mansion in South Beach (Miami) has apparently flipped back and forth between being a public hotel and a private club. It probably has about ten rooms. I have the impression that sometimes the pool is closed for private events.

AmEx used to have access to rooms in a few private clubs, especially back when they had a program giving guest privileges for Platinum/Centurion cardholders, but the list has diminished to almost nothing. Toward the end, I stayed in the St James's Club, London, before it became a public hotel, and was disappointed.

This reminds me of the a semi-private business hotel chain or private club with locations in NYC (Wall Street area) and elsewhere where your company or corporate sponsor (client, etc.) apparently needs a membership in order for you to stay there. Check in is automated and requires a credit card IIRC. It's not luxury but feels like some of the NYC university clubs, although the one near Wall Street has a lively bar at happy hour time and a good restaurant. Not luxury, but some good locations for business trips.

In addition to university clubs, let's not forget private clubs like the Cosmopolitan Club NYC, Cosmos Club and Army and Navy Club in Washington, etc.