As an exercise, does "give unto Ceaser ..." explain why say, the Chinese succumbed (Unequal Treaties, Opium Wars)?

China had in a sense the opposite problem from the Islamic world, no concept of a legitimate institution independent of the central government.

Whatever it was that made the British beat the Chinese back then or makes China ascend so quickly today as to leave All of Europe combined let alone the UK in its dust,

Careful, 20 years does not a historical trend make. The only reason it appears this way is that a European bubble is in the process of collapsing, whereas China's hasn't yet.