Most of the job is exceptionally routine to the point of being boring. In busier airspaces the most common stress is just associated with maintaining timing. Ironically the ground controllers face more of this problem than anyone. The further you get from the airport and it's class B airspace the easier it all gets.

Until an emergency or a conflict suddenly occurs. There's often very little you can do here other than quickly and clearly provide the necessary information and instructions to aid pilots in averting the disaster. The pilot is in full control during emergencies and you're simply there to give them anything they need. In a severe emergency and in an ATC center they're going to dedicate you to the emergency and bring another controller on to manage other planes in that airspace.

As the technology became available to give planes the ability to see and avoid each other with Traffic Advisories and automated Conflict Resolutions we made it mandatory equipment for passenger transports. We made it mandatory for pilots to obey this system with _higher priority_ than any prior or new instructions from ATC.

So you want people who think ahead, constantly prepare for conflicts, and have a reliable level of vigilance. So when the emergency happens they're situationally well prepared and capable of managing all available resources that their stress levels barely increase. A bad weather day with lots of cancelled flights and closed airports should be the highest stress factor they face in their careers.