I wonder how many people in Germany, Italy or the UK could really name our Queen as Mrs. Elizabeth Mountbatten-Winsor. But then it is possibly that the citizens of Germany are more familiar with the Saxe-Coburg dynasty.
The French royal family continues to generate tourist revenues for France, with Versailles being a popular destination. Even the pharohs of Egypt, despite no longer ruling, continue to attract visitors to Cairo.
I doubt that an elected president would work out much cheaper than the royal family does today, just think of the cost of all those pesky elections, not to mention a referendum to decide if the role should cease being hereditary.
But whoever is the head of state, it will not stop tourists visiting the Tower of London, Winsor Castle, Buckingham Palace, and Hampton Court. Royal weddings, funerals, and corinations do not happen regularly enough to make any significant difference to the the 6th largest economy in the world.