Christianity hardly has a perfect history as to its function in politics, if historically speaking, on average, it has worked rather differently in worldy matters, politics included, than has Islam, and the reason is Mecca.
Christ was not a very worldy figure compared to Muhammad, as to leaving a footprint in terms of a political community like Mecca.
Muhammad on the other hand was very much a worldly figure, not only as to wives, but also as to founding an empire.
Along the way, the created questions of governance, which are answered both in the recitations of the Koran and also in what became either convenient legends or in fact statements and actions of Muhammad known as Hadith as to what to do in certain situations.
Thus, Islam from its foundations was very much more a theory of governance than Christianity, which has large implications for how Muslims tend to respond to changes in challenges.
In particular, from the very beginning of Islam, when the Umma, the Mulsim Community, felt a sense of threat, like the rise of European power, it tended very strongly to see that threat in terms of not being good enough Muslims living the way Muhammad did in Mecca.
Thus, with the rise of European power, although some always argued for adapting European ways as the answer to relative Muslim weakness, the logic of Islamic tradition made it more likely that one would try to "purify" the Islamic world towards what had been the traditions at Mecca.
That is what the Muslim Brotherhood and the various other conservative Islamic movements share in common, which imparts a rather different orientation to politics, if there are analogies in China and Japan, if in the end China and Japan and Korea did adapt much more fully to the nature of modern Western civilization, now not really purely Western with the rise of Japan and now Korea.