Excellent "fundamental basis" for authorities and ANSPs post
Thank you Eyrie for clearly reiterating the fundamental reasons for both authorities, and ANSPs.
But there is also one more responsibility for an authority however, and that is to assure a specified level of safety for "Paying passengers", who otherwise buy a ticket in "common Carriage" (in the US this is often called the "Highest level of safety" dating from the original FAA Act, which in itself has always been problematic language, since the highest level of safety is literally to not fly at all.
For ANSPs, there is also a typical responsibility to provide agreed facilities (to the extent that they are wanted by the airspace users, and the users are willing to pay for them - e.g., navaids, lighting systems) to facilitate flight operations.
The global aviation system needs to review and get back to, and better understand these basic fundamentals, if there is to be any hope of GA economically and practically surviving for another century (and the same for other airspace users to survive too).