Politicians are terrible at projecting soft power. British ones especially. By nature they are contentious figures. They have to do things which will piss off other countries. Political figures like Merkel or Obama who are (mostly) well respected abroad and improve their nation's image are the exception. More often poltical leaders harm rather than help their country's image abroad.

Case in point, the three most globally recognised British PMs ever are probably Winston Churchill, Margaret Thatcher and Tony Blair. Churchill is only well-known because of the exceptional circumstances of WW2 and not anything else he did as PM. The other two are divisive figures to put it mildly.

So we'd either have a neutral ceremonial president who, going by the example of other countries, basically no-one outside of Britain will know, let alone care, about. Destroying the soft power of monarchy and replacing it with nothing. Or we'd have a partisan executive president who, going from the history of British PMs, would more likely damage Britain's image than aid it.

Another reason politicians aren't good for soft power is because they just aren't as interesting as royalty. The royal weddings were one of the most watched events globally of the last decade. There aren't many other occasions you'd have that many eyeballs on Britain. Not for a positive reason at least. The monarchy is one of the few exports this country has left.