I'm having a hard time understanding the genesis of this interpretation of history. I think I have a basic understanding of how the Mongol's took over the country and commandeered the existing bureaucracy, but I don't understand why people believe the 'Yuan Dynasty' to be one in a long line of Chinese dynasties as opposed to a foreign take over.
I especially have difficulty reconciling this given 1) the rancor towards the Dzungar Khanate (a genocide) who traced their lineage to the Khans and 2) that other foreign conquerors who took over parts of China are not remembered fondly. Any recommendations for further reading would be greatly appreciated.
(I don't think this counts as current events but I'm assuming this belief is older than 20 years since I've heard it from older people.)
If I understood your question correctly, I think the short answer is related to the contemporary view on the legitimacy of the invader dynasties such as the Qing.
The Manchu emperors were the one who first emphasised "China" as an early form of nation state after their initial conquest (Mainly to legitimise their rules). At the same time, territories such as Xinjiang, Inner Mongolia and Tibet were brought under formal control or influenced by the Manchu imperial government.
Now, when the Chinese Nationalists (the majority of them were ethnic Han) overthrew the Manchu government to establish the Chinese republic during the early 20th century, there was debate as whether these non-Han regions should be considered as part of the new nation state of China since they were incorporated into the empire by the invading Manchu.
The debate was not really resolved due to the chaotic Warlord period and World War II. By the time the Chinese Communist takeover, they emphasised state power over ethnic boundaries and re-asserted controls on the non-Han regions according to the frameworks established by the Qing. In doing that, the current mainland Chinese government would naturally acknowledge the Qing dynasty as a "Chinese state".
Back to your question. If the Qing dynasty was considered to be "Chinese", than the Yuan dynasty would also be qualified since they shared similar lineage and circumstance. I think this is at least the "official" viewpoint in mainland China.
It is also worth knowing that the Yuan at the time had no problems asserting their rules in China in reference to the previous dynasties and styled themselves as "Emperor of China", while at the same time claimed the Mongol Khan title.