There is a weak spot in having an elected Head of State with any power at all. Being attractive to the electorate is no guarantee of being impartial or even honest. Someone with charisma but no scruples could charm his way into getting elected and then use their position and influence to get more and more powers.
The power to veto a bad law? Who defines a bad law? The HoS?
The power to refuse to sign an international treaty with certain countries or on certain subjects ? Who defines which countries are not suitable? The HoS?
The power to fire a bad PM? Who defines a bad PM? The HoS?
The power to disolve Paliament against its wishes, before its set time is up? Who defines a bad Parliament? The HoS?
Who is making the big decisions, the HoS as one person with one person's opinion (however many learned advisors they have), or an elected Parliament? who have access to the same number of learned advisors?
What we have in our "old-fashioned, anachronistic" monarchy is a position as Head of State which has been moulded and whittled down for over a thousand years from all-powerful dictator who ruled by virtue of military conquest and whose word was law. This was done through repeated interventions to reduce that power and share it.
First religious leaders pushed God's approval. The king granted them land and rights, and also to warriors in exchange for military support. Then those same barons, once they had made enough money out of their lands to become money-lenders to the King, wanted more input in exchange for funding his wars and the costs of policing his kingdom, and for interceding with God on his behalf.
The talking-shop where they advised him on policy became Parliament. Gradually more sections of the population fought their way into being represented at that talking-shop - the common people as well as the lords, and eventually women as well as men. The leaders in the talking-shops became leaders in the country.
Each addition of more voices to the decisions diluted the power of a single individual until what we have now is a Head of State who, in practical terms has no formal personal power . Grany and some others think that is a bad thing, and want the HoS to have the supreme power of over-riding Parliament (which is the embodiment of the whole population) by completely sacking the PM and Cabinet and hence contradicting the lot of them.
That is a LOT of power. It would be better to strengthen the power of the whole Commons and Lords to be able to get rid of a rogue PM and cabinet from within and to reduce the power of the Whips to force individual MPS to vote as instructed instead of using their own brains and the knowledge of what their constituants had voted them in for.
We say that Parliament is sovereign, but there have been efforts made by an individual to reverse that sovereigny and make the post of Prime Minister approach that of Monarch-as-dictator. President-as-dictator could be similarly subverted if they could over-ride Parliament by their sole action.