there's something to be said about separating the head of state from from the head of government. The Queen was mostly a unifying figure (well, maybe not for The Sex Pistols, but for most of the UK). There are certain events, like honoring war heroes and marking significant cultural events, that should be unifying and observed by the Head of State. Saluting the national soccer and Olympic teams or honoring the winners of a NCAA championship, perhaps the Kennedy Center honors too. I always feel weird when a pol uses these events for electoral gain; it seems divisive. I could go into examples, but that's for another part of the board.
They are in some respects separate issues. There are basically three structures for democratic governance - the parliamentary system where the head of state and head of government are separate and the head of state is basically ceremonial/tradiotional, the American-style Presidential system where they are combined, and the French-style hybrid system where the head of state and head of government are separate, but each have actual power. I think De Gaulle's idea was that the President would represent matters of concern to all Frenchmen, and have actual power in foreign and defense policy, while the Premier would act in domestic areas where interests might be more divided. The practical results may be different and better addressed in another part of the board.
But one can have a parliamentary system without a monarchy. The German and Israeli presidents, like the British monarch, have pretty much only ceremonial roles. However, I suspect that even political junkies who can name every Bundeskanzler since Adenauer and every prime minister since Golda Meir would have a hard time naming more than a handful of presidents. So while they do have that ceremonial role, they don't have the strong cultural, symbolic and unifying role that a monarch has.
A bit of trivia: one of Queen Elizabeth's cousins got to be both head of state and head of government. Simeon of Saxe-Coburg-Gotha was Tsar of Bulgaria from 1943 to 1946 and Prime Minister of the Republic of Bulgaria from 2001 to 2005.
