That is not entirely true.
In 1260, when Khublai became Khan, he conferred upon Phagpa the rank of Kuo-shin, the State Instructor. Later, after his nirvana he was promoted to the rank of Ti-shih, the Imperial Tutor. Although at that time Tibetan Buddhism was only the religious faith of the Mongolian imperial clan and the people of the higher class, it had begun to greatly influence the thought of many Mongols.(snip)
Be that as it may, as already mentioned, the Buddhism espoused by Khublai Khan and Phagpa, The Great Master of Sakya, flourished only among the Mongolian ruling class, failing to profoundly affect the masses. Consequently, following the collapse of the Yuan Dynasty (1368) and the isolation of Mongolian lands, Buddhism faded away along with other foreign religions, giving way to the revival of the old shamanistic religion.
More about Mongolia can be learned here.
Kind of my point. They weren't mostly or primarily Buddhist at the time of their great conquests. Buddhism, Christianity, Islam, Manichaeism and probably Taoism and other religions were practiced freely by Mongols. This is especially because the Mongols largely absorbed rather than destroying other nomadic tribes. There were a great many more "Mongols" by the time of Kublai than when Genghis started his career. Of course the assimilated groups all brought their own religious practices with them.
The Mongol rulers respected and supported all religions, up to the point when the western and central khanates began converting to Islam.