I read that Chinese culture has not been broken for 5,000 years, but I think it was broken during the Yuan Dynasty, is that right?

Whether the culture was broken or not and whether the country was occupied by foreigners or not are two different things, personally. Although the Yuan dynasty suppressed and persecuted the Han Chinese, it was still a Han culture during that time, and the Han Chinese didn't speak Mongolian because of it, and didn't change farming to sheep herding because of it.

To put it another way, is the culture of the minorities included in Chinese culture? If it is, then they were annexed by the Han in the first place; if it is not, then the Han civilization has long been infiltrated by the minorities in several major ethnic blends.

Culture, like lineage, is constantly compatible and evolving. Just as the vast majority of Han Chinese nowadays may contain ethnic minority blood, we can't say that the Han Chinese nowadays are not Han Chinese. The same is true for culture. We cannot say that Chinese culture has been broken because of the penetration of foreign Mongolian culture. For example, when western culture entered China, there were a lot of imported words and cultures, and even a lot of lifestyles have been westernized, but we can't say that this is not Chinese civilization.

I think the key to whether culture is broken depends on the cultural orientation of the majority of the people in a society, not the head of state or the ruling party. For example, if a descendant of a black man becomes president, American civilization will still be American civilization.