In your example, who gets to decide who will use the "private" property? The government? The person who got there first? Does it become a sort of law of the Jun

In your example, who gets to decide who will use the "private" property? The government? The person who got there first? Does it become a sort of law of the Jungle that the person to defend a place is the one who gets to enjoy the view? At least in the latter the property is truly owned by the public. But in the former, you simply made the government the rent seeker instead of a individual person or company. The end result is no different.