The constitutional monarch is a figurehead without much of a day to day role, but who ensures that the demagogue we elect is not at the pinnacle of the political hierarchy.
That's more important than most people understand. A prime minister is always aware of the limits of the role.
There are lots of republics in which the president has only a ceremonial role. Germany and Italy, for example. There are absolute monarchs as well. The title does not indicate the effective power of the role.
> A prime minister is always aware of the limits of the role.
So is the president, if you have an appropriate constitution. This is the key, not the fancy title you give to your head of state.
A monarchy has the downside that who gets to be the monarch is fundamentally undemocratic.
So I get 1/20 millionth of the decision in which of 3 or 4 members of an exclusive elite get a patronage position, and it's all good?
I'd just as soon eliminate the risk of handing that role to a political climber and leave the position to someone whose power we all clearly understand is only formal.
That being said, I understand and appreciate your arguments around a properly framed constitutional role for a president. If we ever do make the change, I'd like a non-imperial presidency...
Ireland is a much better model that would largely keep things as is, by replacing the Governor General with a ceremonial presidential figurehead.
Prime ministers are representatives of the legislature, not above it, and presidents in a normal parliamentary system are like VPs in a US-style system - really just there for tiebreaking (and in that the responsibility for organizing government succession in ambiguous situations, but no real latitude.)
A monarch does nothing but make sure that a country always has race and rule by blood at the center of its constitution.