Canada’s head of state is The Queen of Canada (Queen Elizabeth II), but because she’s rarely here, we also have a Governor General as her national representative and a Lieutenant Governor in each province. In theory, the Queen appoints them on the advice of the Prime Minister, though in practice she just rubber stamps the Prime Minister’s choice. I don’t think electing them would actually give better results.
There’s a benefit of having someone non-partisan in a ceremonial position. Any time the King of a Middle East nation shows up in the US, the media points out that President Obama bowed down to him or President Bush kissed him on the cheeks and held hands. When you have someone appointed who doesn’t have to worry about looking stupid because they don’t have an upcoming election, they’re free to do all that ceremonial bullshit.
