You'd hope an air traffic control centre would have a big box of popsicle sticks and black pen markers to cover the fallover of power | backup power | digital systems.

Well, they do (figuratively). That is why the entire airspace is not totally shut down, but capacity "just" substantially reduced.

I believe that's literally what they fail over to, yes.

Per aircraft physical batons were literally the way a number of pre digital air traffic control systems worked.

Whoever has the stick is responsible for the aircraft - it was a literal "unique token".