I feel like the premise of this question needs to be somewhat qualified. The Yuan and Qing adopted elements of Chinese culture as a matter of outward expression to legitimize their rule among Han subjects, but only adopted more substantive elements of Chinese culture (institutions, writing systems, philosophical / religious customs) in bits and pieces.

Tensions between Manchus and Han Chinese simmered right up until the Republican period, when revolutionary Chinese leaders tried to frame China as an ethnically inclusive nation. The stereotypical queue hairstyle was a very visible Manchu imposition on the Han population. Government, especially at the top levels, was segregated to an extent, where Han Chinese were discriminated against, while people like Mongols were advantaged relative to Han. Parts of the empire were supposedly reserved as Manchu-only preserves to serve as a bastion of last resort in case of a successful Han rebellion. The banner armies eventually came to be staffed mostly with people of Han origin, but these soldiers were actually seen (by other Han) as being Manchu-sized Chinese rather than vice versa. The imperial government used the Manchu language, with its own script, right up to the 19th century.

So, sure, there is a great deal of sinicization going on but the extent to which it occurred I think is exaggerated in some circles.

Because the Mongols and the Manchus were both very small % (something like 1-1.5%) of the Chinese population. What this means is that neither dynasties can hope to displace the native Han population and therefore had to adopt some system of governance to rule over them.

The Chinese system of legitimacy: the Mandate of Heaven, allowed a pretty wide breadth of people to rule China, as long as they obey and follow a set of governance and moral principle roughly corresponding to a philosophy known as Confucianism.

Key parts of Confucian system of governance involved support for and allowing the Chinese intelligentsia to rise up in the governmental bureaucracy through an exam system based on the Confucian classics. This is actually quite advantageous since both the Yuan and the Qing needed them as a source of talent for governing the country.

This is also necessary since the Chinese intelligentsia comes from the landed gentry which de facto controlled the country on the local level. Not allowing them to rise up through the imperial bureaucracy through the exam system amounts to a breach of the social contract between them and whoever controls the central government. And you can't have this because a rebellious/angry gentry would end up making the country ungovernable for the dragon throne.

So basically it was necessary for the Yuan and the Qing to act like Confucian Emperors, find themselves a position within the Chinese dynastic system, and adopt elements of Chinese Confucian culture to govern and control China because that's what made their governments legitimate in the eyes of the literati gentry.

It should also be mentioned that Chinese culture far exceeded the Yuan and the Qing in terms of sophistication and level of development and thus it seems natural for them to assimilate into a more advanced society because it's far more appealing than nomadic/semi-nomadic life style after a generation or two.