This only applies if the driving team is trailing or tied. This is because they benefit from having the clock stop and getting set for the snap much more quickly than if the clock had been running from the last snap. You could say "Well, tough ****, the refs made a bad call and it should still affect the team that almost got screwed by that bad call time-wise"; but I'd hate to lose a Super Bowl in a scenario where, say, Marshawn Lynch seems to break the plane with 5 seconds left on third down, only for the call to be overturned and his team gets a free shot from fourth down when the game should be over unless they had a timeout to call when he fell short.
Had White been ruled short immediately, the Pats would have either called a timeout or tried to get set as quickly as possible for up to 4 attempts from the 1. This rule simulates that.