The argument against the Monarchy (especially in a Constitutional Monarchy like here in the UK) for me always falls apart when the alternatives are considered.
Whenever new elected posts are created (such as mayors, police commissioners and so on) immediately from the woodwork emerges an assortment of slime-balls, time-servers, party hacks, washed up politicians and some who have never been in the sea to get washed up to start with. They all stand on Party bases and see a nice little earner coming their way if they can persuade enough gullible fools to cast a vote for them. If the post of "President of the UK" came on the market it would attract the worst of the worst of these. (I can think of a few names, but it doesn't really matter - in this respect they'd all be as bad as each other).
There would be no point in having an elected President unless he or she assumed some constitutional powers. There would be conflict between the "President" and Parliament, especially if the Commons majority and the President were on opposite sides of the political divide, and the sovereignty of Parliament would be in jeopardy.
The constitutional Monarchy has served the UK perfectly satisfactorily since the Glorious Revolution. It entails no democratic deficiency because the Monarch has no effective powers to propose or amend legislation. In short, it ain't broke so there is absolutely no need to fix it (especially when the cure will be far worse than any imagined disease.
The country has enough (in fact far too many) elected representatives. It needs no more, especially in a post that would attract candidates with delusions of grandeur of the very highest order. The Monarch is apolitical and the system works perfectly well. We do not need to elect our Head of State.