I believe it really depends on the jurisdiction, both state and local. Some cities have special zoning and permitting regulations regarding "SOB's" (sexually-oriented buisnesses) which include not just ABS's but also strip clubs and possibly bathhouses. The "private club" question reminds me of other "vice-related" laws and workarounds that had been used for many years in Utah and here in the "dry" or "semi-arid" counties of Texas.
I live in the Houston area where there are a plethora of strip clubs and ABS's as well as two bathhouses. There used to be a sexclub, too, but it closed, mainly because of the location on a then newly-constructed light-rail line and redevelopment. It had been previously the site of a notoriously sleazy gay bar with sex on-site. As time changed, there was the "image" question about land use along the rail line and pressure from developers. Although Houston has no zoning, market pressures still have their way, not to mention the business licensing questions that affect the other SOBs regarding their location, size, parking, and so on.
It's a complicated question, really. You'd have to ask it with respect to a particular place where you might have a bathhouse or ABS.
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