People who don't know much about GPS think the satellites are eternally autonomous but that is not the case. There are precisely two uplink control points on the planet which control them. Both in the USA as I recall.
Another interesting problem is lifetime mismatch. It would be foolish to create a system which depends on and could outlive the current GPS constellation.
Also being in the USA we tend to assume the fedgov is on our side and our side is everyone's side, but the rest of the world has found out the hard way many times over to never assume that is or will always be the case. So absolute best case is you should never deploy a worldwide system with a SPOF controlled by a foreign power.
"The story here is that we thought we had designed a system which can do this without human interaction"
No it was designed for humans (mostly US soldiers) to geolocate themselves, so they could help orient themselves on paper maps. This whole idea of pasting a navigation system, or even worse, an autonomous navigation system, on top is a good example of feature creep resulting in systemic failure.
You probably could design a redundant, tough, reliable, world wide (or at least, wide range) autonomous navigation system. It would not look anything like GPS. It would probably look a heck of a lot like a weird cross between the VOR system and authenticated LORAN. It would take a lot longer than a HN post to nail it down exactly, but I'm sure that whatever it optimized down to, it wouldn't resemble GPS very closely. In the grand tradition of all copier machines being called Xerox machines, I'm sure that whatever this successor system is called, the general unwashed masses will continue to call it a "GPS" to the immense confusion of people who actually know what they're talking about.
A planetary scale authenticated mostly decentralized web of trust mesh network of millions of stationary beacons and mobile inertial navigation systems with computational countermeasures to fight intentional bad actors? Probably providing global internet access via the mesh while you're at it, because it's there?
