They were thinking dark ages all over again - a repeat of what Attila the Hun did to Europe. The Mongol empire of then was arguably the greatest empire in history taking in land at the speed of their horses. They probably would have taken over Europe if YG didn't kill the great Khan. (or was that a later Khan?) Nothing short of a total replacement of the government (or 2 miracles as for the Japanese) would have saved Song.
Wasn't Song one of the last empires to fall to the Mongols?