By barbarianism, I'm just assuming you mean Mongolian cultural practices. Well, first consider that only two times in China's history has it been under non-ethnic Chinese (Han Chinese) rule: The Yuan dynasty with the Mongolians and the Qing dynasty with the Manchurians. Of those, the Yuan dynasty lasted only 100 years. Not very long in the span of a ~4000 year old civilization.

And when I say that China was under foreign rule, it most definitely was an occupation, but only successful militarily. Because of the vast rural agricultural nature of China and the very large population, it was a nightmare for the Khanate to actually rule over the area. The population itself was divided into four major groups: the Mongols, the "Central Asian Auxiliaries" (Naimans, Uighurs, Turkish peoples, and Tanguts), North Chinese, and South Chinese. The Mongols themselves remained the ruling class, while the Auxiliaries took over the roles of government workers, soldiers, and merchants. Additionally, since the Mongolians were often not literate in Chinese, much translation work was taken up by the Uighurs in bilingual government offices. This language separation and its effective distancing between the ruling class and the majority of the population is one major point of why the ethnic Chinese never really inherited anything from their rulers.

Another problem is that the Southern Chinese especially were strictly forbidden from engaging with many aspects of society under Mongolian rule. They were forbidden to marry Mongolians, carry arms, and at certain points forbidden from learning Mongolian or any other foreign language. It was a true occupation in the ethnic Chinese eyes, and as such the only real thing of note that is still upheld in China is some of the poetry from the era.