I've written about this before, many times, but it is possible to analyze the subject from the opposite assumption that the political system is a given and Christianity must fit to it - following which the situation will, of course, change again.
But perhaps the political situation can be put first.
So, when the Roman Empire Christianized, there was already an Emperor; it is made sense that the Emperor included Christianity seamlessly into the imperial administrative system: the 'Byzantine' polity, associated with Eastern Orthodoxy and later transplanted to Russia.
This is not a pure theocracy, since the Emperor/ Tsar is not the chief priest/ head of church (indeed, nor a Priest of any kind) - but seen as an Apostle and Christ's representative (Vicegerent) on Earth: the spiritual and temporal leadership roles are split between persons, even though the social system is undivided.
But this fully integrated system has seldom been possible elsewhere, because of weak monarchies dependent on the church, and hostility between a church administered from Rome, and a local monarch - so we get the familiar dual system of semi-separate domains of temporal and spiritual power, in various and changing mixtures of cooperative and competitive, found in Western Europe.
Interestingly, the Byzantine system was almost reintroduced in England with Henry the Eighth and the Church of England - but this became enmeshed in the Reformation and the reorganized CoE became significantly less Catholic.
By and large, Reformed churches have been associated with less powerful monarchies, or republics - situations where the hierarchy of power and spiritual authority are altogether less clear (and indeed hierarchies themselves are continually under critique and challenge).
The political leader was, in devout societies dominated by a Protestant denomination, certainly expected to be a Christian and behave as such - but in general had no power or authority in the church/s.
More recently, albeit only for about one generation, there was the 'pure' theocracy of Brigham Young in Utah/ Deseret - in which the equivalent of chief priest/ head of church was also the supreme political leader: the roles united in one person. This apparently worked very well in its context, but was crushed by the US federal government before it could be seen whether it was a long term viable mode of government.
In all of these systems there are various ways of selecting leaders - spiritual or temporal leaders - and these are conceptualized very differently.
In all, there is an element of God's will being expressed in who becomes a leader - but this can be more or less explicit, and the principle gets interpreted very differently.
In particular, it is an error to suppose that monarchies need to be, or usually are, hereditary; and even when the are hereditary the principle is variously applied with various levels of strictness with respect to succession. Byzantine monarchies, for instance, has no system for succession - the next Emperor was assumed to emerge when needed, and according to the needs and just deserts of the people.
My point is that it seems less and less likely that we can use any of these systems as a blueprint, to be applied generally - since ideas of the ideal political system differ so widely between denominations, and however the ideal is defined, it is but seldom/ never realized.
Actually existing Christianity is always in a severely sub-optimal political situation.
Probably the best rule is for Christianity - as best it may, and constrained by the degree of faith - simply to do what it must do, insofar as it can; and do keep on doing so stubbornly and without accepting dilution or meeting the political world in a compromise.
(Except when something of the sort is literally forced forced upon Christians, when they will drag their feet, and never assent to or agree with the imposed policy.)
In sum, to leave the political system to organize itself around unyielding, devout Christian practice - and see what happens...