Did Yuan dynasty empires ever ruled Tibet as they claim or was it always an independent state as claimed by The Tibetan Govt in Exile?
There is an ancient relationship with Tibet, but it is with the Mongols, not with China.
During the Yuan dynasty, Tibet was part of the Mongol Yuan empire, but not part of Yuan China. They were two entirely separate administrative units, that happened to share one ruler. It is true that the office administering Tibet was located in Beijing, but it was an independent office reporting directly to the Yuan (Mongolian) leadership. The wiki link compares it to the India Office in London during the heyday of the British Empire.
During this period the Yuan (Mongolian) rulers of China converted to Tibetan Buddhism. Slowly this faith edged out their traditional native religion and became the dominant religion among the Mongol people throughout east Asia. (Even today, most Mongolians are Tibetan Buddhist.) So Mongolians developed a relationship with Tibet that is somewhat comparable to what happened between Europeans and the Papal State during the Middle Ages. Whoever happened to be the biggest military power in the Mongol World would seek to legitimate themselves as Khan by being the "patron" of Tibet.
On the ground in Tibet the military forces enforcing Yuan rule were Mongolian, not Chinese. When the Yuan dynasty fell in China, so did Yuan rule in Tibet. For the next four centuries, to the extent there was foreign influence in Tibet, it was Mongolian. During the latter half of this period, the Dalai Lamas were effectively clients of the Mongols.
The first non-Mongolian outside rulers of Tibet were the (Manchurian) Qing starting in 1720. However, they also converted to Tibetian Buhdisim and attempted to portray themselves as just the latest powerful foreign patrons of the native religion.
The first instance I can find of Tibet being ruled for a significant period of time as just another Chinese province by ethnic Chinese who were not by and large Tibetan Buddhist was after the PRC invasion in 1950.
There is an article on Tibet on Wikipedia.
Emerging with control over most of mainland China after the Chinese Civil War, the People's Republic of China incorporated Tibet in 1950 and negotiated the Seventeen Point Agreement with the newly enthroned 14th Dalai Lama's government, affirming the People's Republic of China's sovereignty but granting the area autonomy.
The reasons could be:
In 747, the hold of Tibet was loosened by the campaign of general Gao Xianzhi, who tried to re-open the direct communications between Central Asia and Kashmir. By 750 the Tibetans had lost almost all of their central Asian possessions to the Chinese. (...)
The Qing dynasty placed Amdo under their control in 1724, and incorporated eastern Kham into neighbouring Chinese provinces in 1728. (...)
There is also another article on Wikipedia, which - I think - answers your question:
The position of the People's Republic of China (PRC), which has ruled mainland China since 1949, as well as the official position of the Republic of China (ROC), which ruled mainland China before 1949 and currently controls Taiwan, is that Tibet has been an indivisible part of China de jure since the Yuan Dynasty of Mongol-ruled China in the 13th century, comparable to other states such as the Kingdom of Dali and the Tangut Empire that were also incorporated into China at the time.
The PRC contends that, according to international law and the Succession of states theory, all subsequent Chinese governments have succeeded the Yuan Dynasty in exercising de jure sovereignty over Tibet, with the PRC having succeeded the ROC as the legitimate government of all China.