Maybe JFK needs to figure out how to manage 4L and 4R on the same frequency. In this case, there was no way for the 3 crews cleared to cross 4L to hear the takeoff clearance, or any way for the departing Swiss flight to know that 3 aircraft were being cleared to cross their departing runway, because they were on different frequencies. I imagine all 4 crews would have picked up on the impending conflict immediately if they had the full picture, but were denied the situational awareness they should have had.
Everyone is very lucky the timing worked out as it did - a few seconds difference and there easily could have been three or even four fully-loaded airliners involved in what would probably have been the worst accident in aviation history!
I hope something significant comes from this investigation - we can't continue to rely on luck and angels forever...