To what extent did the death of Frederick Barbarossa contribute to the failure of the Third Crusade?

He lead a large portion of the German Army during the 3rd Crusade (tens of thousands). His death lead to confusion and high desertion rates amongst his troops. When the crusaders arrived at Acre, there were about 5000 left. After F.B.'s death, the leadership of the third crusade fell to Phillip II and Richard I whose dislike of each other prevented the christian forces from acting cohesively as a group. In other words, Frederick Barbarossa's death lead to a lessening of the number of troops and a less than unified leadership which contributed to the inability of them to recapture the Holy Land.

Philippe Auguste and Richard the Lionheart needn't the Barbarossa's death to prevent the failure of the crusade. The king of France wanted to go France quickly.