"Private property" says to me that the Constitution recognizes private ownership of property. Not use of state owned property.
Your premise is correct. Yet the reality is that you rent the land you supposedly own through payment of property taxes to the state.
That's the heart of the matter, the Bill of Rights recognizes property, but Judge Brown (who I respect) states that what we understand as "property" is only what the sovereign US government allows us to hold.
It seems we grafted a medieval tradition into the legal machinery of this republic. While the consistency of a posited sovereign has obvious practical use, it's also provides the government power to seize what we own or compel us to do what we wouldn't do otherwise.
It's the license to confiscate a guitar on a clerk's suspicion that the headstock is decorated with the shell of an endangered mollusk. It's why cops can raid and trash your home without any obligation to restore what they broke. It's why businesses that burned during a riot can not even afford to have the rubble hauled away, let alone rebuild. Or why your neighbor could regrade his property so that yours is inundated with rainwater. It's why a judge in Washington DC could throw an innocent protestor into prison for 10 years without fear of accountability.
It seems it all refers back to this monarchist residue. If we have to have a sovereign, let's go back to a monarchy; democracy with sovereign power is scary.