> If the FAA had enough people and automated as much of the process as possible it would actually improve safety, because they can take action as soon as it's needed without waiting for overworked staff to get through the backlog first.

You misunderstand the root of the issue: a single failure can lead to the deaths of hundreds of people and massive political blowback. Automation is great! Deploying automation always comes with headaches, and here the headaches would be unacceptable because they could easily lead to hundreds of people dying. Jetliners all already have fully digital datalinks, but clearances are still given over analog radio; pilots have the authority to decline a clearance whenever they feel it necessary. How would automation handle that?

The same administration software would have to work across the entire world, including in countries with different aviation systems and standards. Lots of airlines send jets for maintenance in different countries; they would need to trust this system over paper records.

FYI - the FAA is testing out ATC automation for general aviation, which is much more accident prone and therefore a better candidate than commercial aviation which has a sterling safety record. I think if it works there we might see it in commercial traffic, but it's a long road to get that many 9s.