ATC will be fine to get everyone on the ground safely. There are procedures for everything including ATC itself being down.
I never doubted that. The question is if they could've worked without a ground stop by having ATC give them all info.
No. ATC gets busy enough as is, each plane needs NOTAMs covering their entire route, and if the system is down ATC isn’t getting them either anyways.
If I print the NOTAMs for a 600nm flight, they're going to typically go 4+ pages of NOTAMs that are fairly specific to my flight (just for departure, along route, and destination airports) and 15+ pages if I print all available NOTAMs in a fairly small font.
ATC can get them back down, but no new flights should start.