Our monarchy works. It is also inextricably bound up with our unwritten constitution, system of government, administration, military and established church. Untangling all of that and replacing it with an elected president would be to open a huge can of worms.
Also, the monarchy does not belong to the "efficient" part of government but to the symbolic and sovereign part of it. Government exercises powers derived from the monarch and in his name, in conjunction with Parliament, but the monarch no longer personally exercises those powers. He has a veto over legislation which has never been used by any monarch but is the ultimate safeguard should Parliament try to do something unconstitutional or wholly immoral. He is above politics and can therefore unify the nation. An elected president could not do this. Even if an elected president is not a failed politician (they usually are) the fact that they have been elected makes them a partisan figure, divisive for those who have not elected them.
Our monarchy also exercises considerable soft power on the part of the nation. So many presidents are unknown figures on the world stage - no more than mere functionaries. The King of England is not. He is also our chiefest and most formidable diplomat - as he showed recently on his state visit to Germany.
I have heard some criticisms (not here) but in the comments sections on newspapers that the king looked glum during his coronation. King Charles was required to look solemn during the ceremony. His mother also looked glum if you view the newsreel of her coronation. It was a religious service consecrating a sacerdotal monarchy (one of the few left in the world - most other Kings are simply required to swear a constitutional oath). To smile would have looked inappropriate even mocking. To criticise him for looking glum betrays a misunderstanding of the occasion. He brightened up once he was outside again.
The point to note with the king (and the late queen) is that they understand they don't have to enjoy all aspects of their role. I should think they have good days and bad days on the job like the rest of us. But they have to do it out of duty - they are raised to do it out of duty. Most of the time, when the king is out and about meeting people, he smiles a lot and seems to genuinely enjoy it.
"Duty" not "fun" has always been the most important word in the lexicon of a British monarch. It's notable that republicanism gained most traction under Victoria when she hid herself from public sight and refused to do her duty. Edward VIII had to go mainly because his sense of duty was totally inadequate. Duty is very much out of fashion now - we are expected to prioritise fun and an obsession with the self. The monarch exemplifies and holds up the old concept of duty before self. Woke "thinkers" and millennials lack the depth to understand this.
No doubt an elected president would have a sense of duty. But I'm not convinced it would run as deep. It's ingrained in a king who was raised, from birth, to perform that role and educated in the solemn duties of kingship.
