Only in America, perhaps? 
Where I am**, most things that you would call a “trail” have no operator and are not informal. Somebody (a private individual or a larger public or non-public body) will own the land it runs over; there may (or may not) be some legal requirements that they have to follow to keep it in a certain state and it’s extremely unlikely that they themselves created it.
Land in National Parks here is still mostly privately owned. In some cases the national park authority might own the land - I guess that would be closer to the US case, and some other countries may have land management models close to that. Much more will be owned by charities (e.g. the National Trust), water authorities and private individuals.
If something is an “obviously public (FSVO) path” I wouldn’t tag it as informal, even if it doesn’t have an operator as such, other than the actual path users patching it up to keep it usable. If someone’s tried to bushwhack a shortcut then I absolutely would.
** England; the legal rules also apply to Wales