There are two ways to view property that use the terms "private" and "public." One is in regards to ownership, and the other is in terms of access. They are related in that ownership will tell us who gets to define access.
Ownership is about the right to actions related to the thing in question (house, land, business, copyright, etc.) Ownership is best thought of as bundle of rights (actions) that can be exercised without permission. Access or use of something is one of the set of rights that will be owned by someone.
There are different kinds of owners: Public ownership means that all of the people of the political jurisdiction own the property equally but it is managed by the government of that jurisdiction. The managers set the access rules. This is the "commons" - this public property - like state parks, the capital building, military reservations, etc. For reasons nearly everyone at ROR understands this kind of property should be kept to an absolute minimun, not just because of the "tragedy of the commons" issue, but because it takes that property outside of the competitive free market and reduces the beneficial effects of capitalism to that extent.
If property isn't publicly owned, then it is privately owned. That's because the use of the term "ownership" in a discussion related to "public" versus "private" we are talking about who has control - who holds the right to uncontested actions regarding the thing in question.
Access can be declared to be public or private by the owner. Because ownership and access are separate focuses, we can have private property that the owner grants public access (e.g., a restaurant the owner opens to the public) or a publically owned property that is declared to have private access (e.g., some military bases).
The restaurant that you choose to run in the public sphere, on the public commons, with the benefit of access to a broad market of your peers in this nation living in freedom, is subject to a race neutral CRA.
You speak of a "public sphere" and a "public commons" as if the implied owner of a property, who opens that property to those who might want to be customers has, by being open, given up the right to control access. RoR is in the "public sphere" but the access remains under the control of the private owner. It is exercised in a reasonable fashion here, but there are web sites where the very site itself is irrational and where access is irrationally controlled, but as long as it is the owner directing the access, even if it is irrational and immoral, it is - and should be - legal.
People, by right, discriminate - mentally categorize and choose. The discrimination might be on political terms - discriminating against socialists, or it might be done on racial terms. The initiation of force, threat to initiate force, fraud or theft are the only things that can violate individual rights. And to use anything else to determine when to make things illegal for a person to do, we end up violating rights instead of protecting them.
If someone does not want to serve socialists in an internet forum, or does not want to serve blacks in a restaurant, both are individual discriminations that don't initiate force, fraud or theft and despite the fact that one example is rational and moral and the other is irrational and immoral, they both should be legal. We can't make one legal and the other illegal without throwing out the principle of individual rights as the basis of law.
If the government forced Joe to allow socialists to have equal time on RoR, or forced some bigoted diner owner to serve blacks, the result would be the same: forced association, violation of property rights, and justifying laws that are not based on individual rights.
Example: clean air laws. Why should third parties be forcefully associated with the commerce of others? State force then, to me, is justified.
We agree on this. No one has the right to dirty the air in ways that harm others. If they want to dirty the air in ways that only effect their private property, sure, go ahead - e.g., smoke in your house or car. But if enough smoke (like from an industrial smoke stack) gets onto my property, that is a forced association - an initiation of force.
Racist nonsense on the public commons; why should others be forcefully associated with shit for brains sensibilities like that? They should feel free ... to keep that in the double wides, back in the trailer parks, banging their toothless sisters. And don't invite me out there. And keep it out of the commons.
When someone opens a business it does not put their property into the "public commons" in the sense that phrase usually means - it does not convert private property into publically owned property. It just means that instead of implicit or explict invitations to access the property are less restrictive. Because someone opens a business to the public doesn't mean that people can come take stuff and not pay. They can't live in somebody's store. They can't rent out the space to a third party or sell the property. They have an implicit permission to visit, during the hours open, under the terms offered, as customers and no more. No shirt, no shoes, no service. No one is forcefully required to associate with racists. To the contrary, when they can exercise their right to deny access to what they own, by right, whether it is their home or their business, on immoral, stupid, irrational racism, they are exposed and with each iota of moral progress made in a society, they are marginalized further. We have seen this steady progression from a state where slave owning was legally protected, even nurtured, to a society where racists are reviled. Racism is eliminate faster when it is not driven underground by bad laws.
...the real slipperyslope is, bending over backwards to define every random mindless act of a complete shit for brains as freedom.
There is a slipperyslope that leads off in every single direction from the principle of individual rights, which can only be violated by the initiation of force, threat to initiate force, fraud or theft. That is what makes free (chosen) versus forced (e.g., government threats) association. Anarchists, Communists, Progressives, Fascists, etc. would all say that advocating for Capitalism is a case of 'complete shit for brains.' We have no leg to stand on in arguing against their 'shit for brains' arguments without the principle of individual rights. There is no such thing as an emotional revulsion, or a level of stupidity so great that it justifies violating the only clear principle governing the use force. If people chose to be as stupid and repulsive as bigots, that is their right as long as they don't violate rights.