The ATC operating at airports, like ground, tower, departure, etc - are all local and simply transmit voice over AM frequencies. They were not affected so flights could land.

Regional/sector ATC (managing flights en route that left the departing airport airspace and have not yet arrived to the arrival airport airspace) are also transmitting in AM but the actual operations are not where the transmitters are (as a lot of them are in the middle of nowhere). The voice/data transmits back and forth from the actual op centers to the transmitters/receivers over the network. Which is what went down.

Actually some short flights (like flight schools or private pilots on short flights) were allowed to continue to operate, as they never left the airport or arrived at an airport so close that they never needed to talk to ATC managing a sector.