Many countries, including Italy, have laws that prohibit the mapping of military sites, but it is also true that services like Bing, Mapbox, etc., photograph everything. However, OSM requires mappers to carry out GPS surveys to map, so it is not possible to draw the content inside landuse-military in OSM unless it is a soldier at a base, but in this case it would be a crime. In general, I believe that it is correct to map the limits of the military area with landuse=military (just follow them) but I avoid mapping the content (roads, buildings, etc.) visible with Bing, precisely to avoid errors by apps and unwary people. The wiki's disclaimer for military tags simply confirms what is written.
access=private is not the solution because if the map shows a road the user who does not use routing will follow it. Also, there are apps like OsmAnd (https://osmand.net/docs/user/navigation/routing/car-based-routing) with access=private road routing.