A Christian monarch is chosen by a Christian people: chosen, that is, by 'acclaim', which is spiritual, and not by vote-counting election nor by any such formal process.
This is why there cannot be a fully Christian monarch without a Christian people: there cannot, for example, be a fully Christian monarch in England now.
Thus, monarchy ought not to be hereditary as a principle, although sometimes there may be an hereditary succession.
One sign of a good Christian monarch is that they do a good job of securing their succession - they ensure (if possible) that there is a clear and suitable candidate for their own succession.
By this criterion, many monarchs have failed - either they put forward an unsuitable candidate for their own succession, e.g. merely because he happens to be their eldest son; or else the succession is left uncertain and contested.
By this account, our present Queen appears also to have failed, since - although she is indeed as exemplary a Christian monarch is is attainable under the present nonsensical pseudo-system - the default candidate for succession, her eldest son Prince Charles, is clearly not suitable, and would be rejected by the disclaim of any such real Christian monarchists as may remain.
In England, the possibility of a good Christian monarchy was destroyed by the devout puritans embracing republicanism and after the civil war committing regicide (of the martyr King Charles the First), then creating a republic and imposing a dictator (Oliver Cromwell).
In response, after the Restoration the Stuart dynasty enforced a strict hereditary principle of succession, by imposing an unsuitable monarch (King James II) who was rejected by disclaim, and which led to the degeneration of British government into the oxymoronic concept of 'constitutional monarchy'.
The monarchy of England is now fatally impaired, and we are therefore condemned to live under un-Christian forms of de facto republican government - which have devolved to become ever-more aggressively anti-Christian.
So, the impossibility of a truly Christian government is one of the fundamental difficulties under which Christians must labour in these end times.
The consequence does not, of course, prevent salvation; but limits sanctification - and renders the earthly Christian life necessarily less complete than at some points in history, lacking in its potential fullness.
This is something that modern Western Christians simply must accept and work-around as best we may; and if we try to short-cut to a fuller Christian life by trying to impose a real Christian monarch on an un-Christian people, it will merely turn out to be a short-cut to Antichrist.