I favour State Christianity - on something approximating the Eastern Roman Empire/ Byzantine/ Orthodox/ Holy Russian model - and in this often find myself in disagreement with both Protestants and Roman Catholics - and with some statements of CS Lewis (who, at times, wrote against 'theocracy' as the very worst form of government).
But I believe this disagreement is due to misunderstanding of how State Christianity ought to work.
State Christianity is not about forced conversions nor even compulsory devotions - at least, these are certainly not the essence of the matter.
State Christianity is based on the accurate recognition that there is no such thing as neutrality: the State will either favour Christianity, or it will favour something else (another religion, atheism, communism or whatever).
What does it mean, then, to 'favour' Christianity?
To understand, look at modern England, or the USA or any similar nation in the Anglosphere or Europe. Nowadays the State favours the secular perspective, such that all analysis in public discourse, all criticisms and all justifications of policy are done in secular terms (usually 'utilitarian' - that is aiming at increasing happiness or reducing suffering - never mentioning salvation).
Consequently, wherever a modern person may turn, and whatever he may do, he will be confronted with communications and discourses that exclude, conflict-with or deny Christianity.
This happens at schools and colleges, in commercial interactions, from government agencies, and of course the mass media world of advertising and entertainment.
Simply walking around the corridors and streets, he will be saturated with the secular perspective - his emotions and passions will be stimulated; his motivations will be played-upon and manipulated in a manner purely secular, subversive-of and excluding the Christian.
By contrast, imagine all this in reverse.
A world where all of these sources of communications and discourses tended to be consistent with, or to support, Christianity - so that wherever they turn they will be reminded of Christianity, and familiarised with, schooled in, trained in Christian modes of discourse and evaluation.
Where the creative and brainy people would be paid (not forced, or no more than they are now forced) to put their best efforts into Christian-supportive work, rather than as now enlisted to destroy Christianity directly and indirectly. Where artists, musicians and architects would be expected to create beauty, not sickening ugliness and soul-killing emptiness; where writers would be expected to be truthful - not bureaucratically-forced to lie in the service of profits and power.
That is what the State can do for Christianity. Support a Christian-friendly milieu rather than (as now) supporting a Christian-hostile milieu.
So long as the State is a secular realm, then of course the State will support a secular perspective - therefore the State and the Church must ideally work together.
Or, to put it the other way around, if we actually aim at separation of Church and State, as many modern Christians do, then we are aiming at the creation of a large, perhaps the largest, realm of modern life as independent of, autonomous from, Christianity - and this is exactly what we get.
Then the secular State will (as institutions do) grow and extend its sway, encroaching upon the Church, until everything is secular.
In Britain we have reached that point.
How this is specifically arranged is a secondary matter - and of course any possible arrangements will be corrupted because humans are corrupt - but that is the proper aim of society: a harmony of state and Church, with no secular realm.
Note: I suppose it is necessary to clarify that a Christian State is something which emerges from an already-Christian population, and then encourages the process to continue. The Christian State is not something parachuted in, landed on top of a secular society. When the Roman Empire became Christian under Constantine, it was a recognition of the state of affairs and the dominant trends, more than an imposition: but once the State had been Christianized then this led to greater devoutness and a higher state of Christianity (in the Eastern Empire, especially) than had been possible before. Or, to put it briefly, when Christianity was under an anti-Christian state the Saints were martyrs - who died in and for the faith; under a Christian State the Saints were characterized by advanced ascetic holiness and wisdom.