Suppose we created an official state church. Would this help increase religious faith and/or the intensity of belief? Or would the government botch the job of proselytizing, as it does so many others, causing people generally to lose their religious beliefs?
There are many different ways to have an official state church. In other words, there are many different ways one can make an official state church, and we should not assume that they will have the same results. In some establishments, the church is regarded as an independent entity officially recognized by the government for certain purposes. In other establishments, the church is regarded as a government agency or department. In all establishments the implications of recognition are limited, i.e., the recognition does not imply that the State can do anything it pleases to support the church, nor that the State is committed to do anything it can to support the church; but what those limitations are can vary considerably. In one establishment, recognition may mean nothing other than that it receives tax money; in another, it may mean that the church has some limited legislative and judicial jurisdiction recognized as authoritative by the state. Some churches are made state churches to the exclusion of any others; sometimes several churches can be made official. And so forth. So I'm inclined to think the question is unanswerable without determining what kind of establishment is in question, and what the character of the church established is. Gallican Catholicism had relations to the state that were radically different from those had by the Catholic Church in late eighteenth-century Quebec; neither could possibly be confused with the modern Chinese Patriotic Catholic Association; and none of the three are all that much like the Church of England in their relations to the government. And these are all cases where the internal structure and politics of the church is at least vaguely analogous; if we consider other possible forms of establishment, such as one would get by making, say, a Baptist Convention a state church, the relations would have to be yet more different. And it is these relations that would have whatever effect is to be had on the role of religion in that society. So the type of church, and the type of establishment, are both important for answering this question.