I think this is correct, but there is a difference even so. Islam was something like a state religion from a very early point in its history, and there is much in the Qur'an and hadith that reflects this. Christianity got started among mostly-poor mostly-powerless subjects of the Roman Empire, and the New Testament reflects this. It's only hundreds of years later that it became a state religion.
I suspect (but don't know) that this makes it easier to be a reasonably conventional Christian without feeling that your religion should be in control of the state, than to be a reasonably conventional Muslim without feeling that your religion should be in control of the state.
Islam was something like a state religion from a very early point in its history
Not "something like", but "a classical full-blown dialed-to-eleven" state religion. Note, for example, that early Islam knows no distinction between religious laws and state laws. The idea that they could be different would be treated as an idiocy.
This topic is vague and open-ended. I'm leaving it that way deliberately. Perhaps some interesting, better defined topics will grow out of it. Or perhaps it's too far afield from the concept of less wrong cognition to be of interest here. So I view this topic as exploratory rather than as an attempt to solve a specific problem.
What useful purposes does religion serve? Are any of these purposes non-supernaturalistic in nature? What is success for a religion and what elements of a religion tend to cause it to become successful? How would you design a "rational religion", if such an entity is possible? How and why would a religion with that design become successful and serve a useful purpose? What are the relationships between aspects of a religion, and outcomes involving that religion? For example, Catholicism discourages birth control. Lack of birth control encourages higher birthrates among Catholics. This encourages there to be a larger number of Catholics in the next generation than would otherwise be the case, Surely there are other relationships like this? How do aspects of religion cause them to evolve differently over time?