If the power grids went down around the world would all airlines have to ground their planes? Do controllers depend on them to do their jobs?

The most important requirement would be maintaining supplies to the control-towers and the runway lights for long enough for all aircraft to land safely.

The Air Traffic Controllers certainly do depend on reliable electricity supplies, normally the public utility service but from emergency generators if these fail. They need them for radio communications with the pilots, for the radar, and simplest of all, for the control-desk lighting.

It doesn't really matter if the power-cut means no-one can obtain boarding-passes or use the duty-free shops - that's just hard luck. Safety comes first and if the ATC system is out of action no-one is going anywhere, not by flying out anyway.