>The American eighteenth-century enlightenment approach is an attempt to make everything tidy: it's based on the rationalist idea that a thing is its definition and nothing more. So private property is private, that means nobody else can use it: case closed.

You may be unaware, but the American legal system allows for property owners to be held civilly responsible for the actions of uninvited individuals, including criminals with intent beyond simple trepass, that harm themselves on said private property, unless the owner has taken many somewhat onerous steps to post No Trespassing signs often with requirements of details on the signs and posted in short intervals.

So why would a property owner want to allow random individuals to cross their land if it may mean someone can sue them for damages because they tripped and broke an arm, etc.?

The reason sane property owners maintain those roads is for firefighters.

Every state I’ve lived in has landowner indemnification for trails. In Colorado, the landowner must put up signs for hazards, though a local group always exists to help with this. In my experience, the rationale for such a trail to be closed always comes down to a single person ruining it for everyone else.

The second you charge money (for hunting, usually), all that protection goes away.

IANL. Under American law, if the owner doesn't enforce exclusion the land can become public through implied dedication if the public continues to use it over time.