Guess what, all the roads around me are private easements, all privately owned, and they are that way 90% to town. A good portion of my trips never touch a publicly owned road yet I'm still required to display my plate on them. We don't even have public, tax maintained roads where I live (I literally have to bring out a tractor and fix them myself when they wear down). Yet the compelled 'speech' of displaying the license plate is required even then while driving your car on your privately owned non-gated road.
Many farmers have plateless farm trucks, people who live in the woods have plateless UTVs that they drive on private dirt and gravel roads, etc.