Everyone is calm because it isn't a stressful event. Plenty of buffer is built in to ensure that a hazard is not created even when one of the airplanes doesn't do exactly as the controller is expecting.
In the SFO case using intersecting runways, the departing aircraft must clear the intersection before the arriving aircraft reaches the threshold (the beginning of their runway). At this point, the aircraft are still quite a distance apart. When it isn't going to work out, the arriving aircraft is sent around. This puts them several hundred feet, and climbing, above the departing aircraft which likely hasn't quite lifted off yet as he crosses the intersection.
SFO gets a lot more efficiency running these intersecting operations than they lose from the occasional go-around.