The existence of the Vatican state today, and that of the Holy Roman Empire in Medieval times, I think proves you wrong.
No, I don't think they do. Vatican is a state only in the name, and as the famous quip about the Holy Roman Empire goes, it was neither Holy, nor Roman, nor an Empire :-)
The relationship between Christianity and state power is a complicated one and not reducible to a yes/no question.
I also think that if you replace the wiggly "impose itself in law-making and state politics" with much more direct "seek to rule, that is, solely control the state power", you'll find that Islam and Christianity are quite different in that respect. The notion that Islam and Christianity have essentially same attitude towards state is nonsense.
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?