Yes, Canada does. The Queen is Head of State, who appoints a governor general to rule in her place. Effectively though, the Queen appoints a governor general on the advice of the prime minister. The Governor General and the Queen are constitutionally the "Crown" in Canada.
The Governor General is mostly ceremonial, but is mostly a case of Chesterton's Fence. The Crown is an important safety value for when things are abnormal. For example, the Governor General of Canada in 2008 essentially chose who would be prime minister (See https://en.wikipedia.org/wiki/2008%E2%80%932009_Canadian_par...). The issue that arose was that, during a hung parliament, the opposition announced that they would vote non-confidence in the government at the next opportunity and intended to form a coalition. The prime minister at the time instead went to the Governor General to request prorogation, ending the parliamentary session.
This was a crisis that required judgement by the governor general. The prime minister clearly no longer held the confidence of the House, but not officially. On the other hand, the governor general is constitutionally required to follow the advice of the prime minister. She chose to follow the prime minister and prorogue Parliament for better or for worse. I'm not a legal scholar and I think the Governor General got it right, but having an independent individual make these decisions is a feature, not a bug.
The Governor General is mostly ceremonial, but is mostly a case of Chesterton's Fence. The Crown is an important safety value for when things are abnormal. For example, the Governor General of Canada in 2008 essentially chose who would be prime minister (See https://en.wikipedia.org/wiki/2008%E2%80%932009_Canadian_par...). The issue that arose was that, during a hung parliament, the opposition announced that they would vote non-confidence in the government at the next opportunity and intended to form a coalition. The prime minister at the time instead went to the Governor General to request prorogation, ending the parliamentary session.
This was a crisis that required judgement by the governor general. The prime minister clearly no longer held the confidence of the House, but not officially. On the other hand, the governor general is constitutionally required to follow the advice of the prime minister. She chose to follow the prime minister and prorogue Parliament for better or for worse. I'm not a legal scholar and I think the Governor General got it right, but having an independent individual make these decisions is a feature, not a bug.