I think there's a useful discussion to be had about what private property looks like. Should a corporation be allowed to hold a monopoly over millions of acres of land over a span of hundreds of years, for instance? I think most folks, anarchists and extreme leftists included, would consider a home and a family garden a perfectly legitimate piece of "private property". But, somewhere, as the stakes get higher, and the abstractions for the entity that "owns" the property (and who is responsible for what happens on that property, such as pollution, violence, etc.) becomes more diffuse, there is useful discussion to be had about whether that is legitimate private property or not.

I believe it is a stretch to say that an entity as diffuse as the United States (or any nation state) should be able to prevent billions of people from traveling freely from Mexico to Canada.

In short, I think "private property" is a good and useful fiction that we can probably all agree on, when we are speaking of individual humans who live and work on a piece of property. When you begin to make up fictional entities to "own" the property, I am less confident of the rightness of the theory.

Yes, well done. My one-acre homestead is indeed exactly the same as a sovereign state, it in fact changes nothing when the scale is millions of times that large. Also access to private property and state immigration are in fact the exact same concept as well.