I agree that it would be tough to hammer out and get wide agreement upon an alternative justification for exercising authority in the territory now called Canada. "The authority vested in the Crown" - what is the source of that authority? The Medieval European idea of the Divine Right of monarchs to rule on earth? How did the original monarchs come to be? By military prowess and political acumen, by being able to secure and hold power? This is the origin of the authority to rule over the "dominion" of Canada. Some adventurers came over and planted a stick with a piece of fabric on the shore of the continent in the name of these monarchs. Trappers eager for beaver and bible thumpers followed. European technology and alcohol helped subjugate the original population. Skirmishes and negotiations occurred with those other European settlers who rejected monarchy and argued that the right to rule came from the people, at least the wealthy white male ones. Borders were more or less set and new influxes of people were selectively allowed in to labor on and populate the land.
But more and more authority was slowly transferred to those who lived in the colony. In 1982 the Constitution was 'patriated,' a Charter of Rights and Freedoms, which does not refer to any Divine Rights, was adopted, and Canadians agreed on how to amend the Constitution without the monarch exercising any real power over the process. Symbolic power perhaps, but no real power. Historically the monarchy was at the "core" of Canada's constitution, but it's arguable to say that it remains so. We've come a long way since Runnymede, and are pretty much a republic in practice, with the GG and the monarchy called on to decide on affairs only in extraordinary circumstances.
Had Michaelle Jean said no to Harper's prorogation, and had Harper then gone to QE2, the only right thing for Liz to say would have been "You live with what your GG decided."
But there is less mobility up and down social classes if we accept a hereditary elite.