the problem is; you are using the term public v. private in terms of access, which is also how the term was used in the opinions you cite. I am not arguing that point.
The point is my statement was dealing with public v. private ownership. For some reason, in your fervor to engage me in a dispute, you seemed to have ignored that. Instead, you jumped to your "public establishment" as being defined as being synonymous with an establishment, whether publicly or privately owned, that provides accommodations to the public.The establishment in question is a private establishment in terms of ownership and due to that, they are allowed many rights a public (as in "publicly owned") establishment is not necessarily allowed to do. A public establishment is bound by rules and laws that cannot be imposed upon a private establishment regardless of whether it provides public accommodations or not.
so, you can have a private or public establishment that provides accommodations to the general public. Each of those must abide by the civil rights laws in place. You can also have private or public establishments that do not provide accommodations to the general public. The private establishment is not bound by those same civil rights laws, in general, while the public establishment is still bound by those laws, in general, as there really is no such thing as a private publicly owned establishment. Due to the ownership by the public, even if the use is restricted, they still do not have the right to discriminate as stated within the civil rights laws in place.