I believe it's simple enough to establish which operations are truly private.
That's the crux of this whole thing, really.
Two of the best examples I can think of are so far apart that they wind up coming back together again: country clubs and gay bathhouses.
It's been established many times that a country club can openly discriminate as regards who can and who cannot be accepted for membership, with any number of racial/ethnic minorities as well as gender being considered acceptable grounds for criteria either for acceptance or refusal.
Though much less likely to exclude potential membership based on racial/ethnic characteristics, it's clear that gay bathhouses obviously exclude women from any consideration of membership, and for very good reason.
In each case, annual membership dues are a deciding issue in how such establishments can be labeled private. And this follows through even so far as to whom a member can bring in as a guest.
There is also the issue of dress codes, where fully-public establishments can discriminate as to who may enter (whether it be the entire club or just restrictions to certain areas). This is pretty much universally recognized as valid, though it's not only highly discriminatory but also at the subjective whims of whoever is deciding to what extent one can pass or fail, gaining admittance or not. Try getting a table at a Morgan's steak house in a bathing suit and flip flops sometime.
Then there's the issue of condo and co-op boards (especially in Manhattan, but everywhere, really). Many are notoriously tough as regards who will and who will not be considered eligible for consideration for the possibility of purchasing and/or leasing a home in their domain: show business types being the most publicized of those refused.
On the flip side of this are zoning restrictions that regulate everything from density and minimum-sized lot restrictions (1/2 acre or larger, for instance) to style of home built in privately-developed, gated-community type areas as well as more open, more "public" real estate schemes.
There's at least one town in Massachusetts (
Hingham), for instance, which mandates only white Christmas lights be used: nothing blinking and no colors. I've always thought of this as an example of absurd snobbery but hardly the oppressive boot heel of a police state.
Historical districts are also notorious for the expensive and time-consuming restrictions on everything from replacement windows to the color you can paint your front door. Anyone wishing to live or operate a business in one knows going in that the rules are narrow and strictly enforced.
And, of course, except for Hyannis, no chain-stores or restaurants are allowed on Cape Cod. This might seem like an odious restriction to free-enterprise marketing, but it has succeeded in making the Cape one of the most unique (and treasured) spots in the entire US.
None of this is really all that controversial. It's all part of real life and is (in my opinion) completely acceptable exceptions to complete accessibility of everyone to anywhere as well as beneficent governmental oversight in an attempt to maintain the unique charms of certain places.