I don’t really think that a head of state is necessarily more closely related to the executive branch than other branches. The British crown is one of the traditional branches of Parliament, correct? (And it isn’t necessary to point out the crown has no real power; almost the same could be said of the House of Lords.) Furthermore, it seems like many heads of state top their countries judicial branches no less than their executive branches.
As refusal has said, there is little that unites heads of state other than “heading” their “states”. Their roles vary as much as the definitions of those words. (The Japanese constitution doesn’t even call the Emperor the head of state - it calls him “the symbol of the unity of the state” or something like that … and moves to change the wording arouse controversy.)
That’s all practice. What about theory? In terms of political theory, the difference between a head of state and a head of government is as important as the difference between state and government. Let’s say, for example, that the government is run almost entirely by socialists, and you are a capitalist. Or maybe the government is full of isolationists, and you are an internationalist. When you run for office, you run against the government.
Is this treason? Heavens no. Treason is when you oppose the state; it is heavily discouraged (it’s the last thing you can be executed for in the UK; other countries frown on it as well). Opposing the government is heartily enouraged - the freedom to do so is part of the definition of democracy.
Look at the term for the second biggest party in the House of Commons: “Her Majesty’s Loyal Opposition” … loyal to the Queen, opposed to the Prime Minister. Countries which fail to properly distinguish between state and government lack something very important: a safe space for the political opposition to occupy.
Most American republics lack a separate head of government. George Bush is indeed head of state and head of government in one person. This can put the opposition in an uncomfortable position, symbolically. When protesters march through the streets carrying anti-government signs, do they shout, “Down with head of government George Bush. Long live head of state George Bush!” Doesn’t exactly roll off the tongue.
I’m not saying anti-government protests are automatically considered treasonous in American republics, but they are considered seditious and disloyal way too often. I seriously believe the symbolism is important - if you could imagine those protesters with a beaming picture of the head of state on their wall next to an old carboard sign saying “The prime minister is a fink!” maybe their marches wouldn’t make their opponents imagine a coup d’etat around every corner.
Some countries try to have it both ways, perhaps. In France, the principle that opposing the state is seditious worms its way into the constitution in the form of a clause which outlaws “offensive criticism” of the President. This would make perfect sense if the Presidency were a benign office charged with baby kissing and red-carpet-for-foreign-dignitaries-rolling. However, the President of France is far from a figurehead. Some say it is a dangerous hybrid, with the Presidency have broad but vague constitutional powers, and the Premier being saddled with all the dirty work.
It is possible to make too much out of the head of state / head of government distinction. I hope I’m not doing that. Americans can distinguish between state and government even if they don’t use that language to do the distinguishing. (A European might say, “Of course I love my state, even if I despise its current government;” an American would say something like “Of course I love my country, even if I despite the current administration.”) I’m just saying: It’s not just executive orders, vetoes, budgets, and chains of command that are important. Symbols are important, and people are important.