What you didn't mention is that without the monarchy, government would be impossible without a from the ground up rebuilding of our constitution.
It is the institution of the Crown, not the person of the monarch, that is the fount of all authority. The Crown is the source of power for the Courts and police, Civil Servants are employed by it and the amred forces hold allegiance to it. Parliaments authority to make laws comes from the Crown, and it is the Crown (via the monarch) that confirms them as the law of the land. Parliaments' mandate comes from the electorate, but not its authority.
So in point of fact, if the Royal Family become too unpopular, it is entirely possible to replace them without abolishing the monarchy. For instance, there are living heirs to both the Stuart and Plantagenet lines. It might even be possible to trace, with modern genetics, descendants of Aethelstan, the first King of the English.
All that said, there is of course, precedent for an elected monarch (many Saxon, Viking and Celtic monarchies were of this type in the past). How this would be worked out in terms of who could vote and who could stand I don't know, but at least it must still have the advantage we currently enjoy of an apolitical Head of State.