I understand the way that it operates from the legal construction. Walmart would be stupid to exclude people based on race or sexual preference, as would anybody else.
Ownership of property or body isn't ambiguous however. It establishes that the owner controls the property or the physical body or it isn't ownership. That's why prohibition of substances is a form of enslavement as it removes an essential element of ownership, which is control.
I try not to frame this issue in should a white shop owner be allowed to exclude/forced to serve a black customer, or the opposite of the same.
A good example might be the following...
Should a Mexican run establishment be permitted to exclude Guatamalans?
One might first answer, what difference does that make? Or then say, no of course not. Or be like you and say well sure if they own the property.
Such things are not good for a society. Although the biggest impediment to our society might come from "racist whites wanting to keep out blacks," if that is allowed so would other ethnicities excluding all or specific other peoples.
This has a negative impact on commerce. And I'm not talking about the bullshit commerce clause reasoning to end segregation way back when.
It would limit local and state sales taxes. Create massive redundancies in the economy. Serve to further divide the nation, instead of uniting it.
Allowing such a thing has a very bad impact on the whole.