In a further evolution of the role of the monarchy in Canada, I suggest we go from a living monarch to an abstraction; …from owing allegiance (in theory) to a flesh and blood individual as symbolic of the State, to an abstract symbol instead. Such transition may be readily done, by simply declining to recognise the current King’s successor. ..Upon his demise all references to the royal authority would be to ‘the Crown’, an abstraction. All imagery would be that same symbol.
The Governor-General and the Lieutenant~Governors would then be the living embodiments of the Crown. ..Their titles and duties would remain as they are, acting symbolically on behalf of the state, as represented by a symbol instead of a living presence.
The Crown as a symbol would then indicate that Canada retains a monarchical form of government, sans a living monarch. ..The Prime Minister, though elected, nonetheless is not symbolic of the State. ..Better for that symbolic position to be kept ceremonial and not inhere in an elected President, who would inevitably be jostling with the Prime Minister for power and attention.
A natural progression
The first few Governors sent out to the colonies that would become Canada actually did govern, did make decisions with the assistance of advisory councils. At first, each council was appointed by its Governor, who was not bound to accept whatever advice might be offered.
Naturally, a newly arrived Governor (however well briefed before arrival) would be unfamiliar with most local matters and therefore inclined to follow the council’s advice. .But inevitably there were scandals, since the executive members had a strong tendency to benefit from government contracts.
And so an elective body was added, which often was in conflict with the appointed executive council advising the Governor. After much agitation regarding Representative Government that council came to be appointed from among the elected representatives of the populace. ..It then made practical sense to draw them from the political party in control of the elected body. …Yet, the Governor could still veto legislation he did not like, though this became increasingly unusual, then rare, and eventually unthinkable. ..From being active in governing, the Monarch’s representative became a rubber stamp signing into law whatever the legislative body passed.
By way of contrast
The Americans, rather oddly, are still stuck in an 18th century conception of power, and their elected king (there has yet to be a queen) retains the right to veto legislation, exercising that right often. ..There is also the matter of pardoning criminals, which is often highly controversial, especially when it is someone politically connected who receives a pardon.
Even more oddly, recent holders of that office — called the presidency — have presumed a right to order assassinations of enemies of the State, provided these are executed outside the territory of the USA. ..Rather redolent of divine right authority, the king being a channel of the divine, a very outmoded conception. …By and large, the citizenry of the USA are strangely blind to how antiquated all this is. Many feel that what they have is perfection, or near enough, or at least best left as is. ..Why mess with it?
In closing
Today, the King’s representatives (federal and Provincial) have ceremonial duties and little more. And at present it is left up to the Prime Minister of the day to select a Governor~General and the several Lieutenant~-Governors.
However, it would seem prudent to devise some other formal means of making these selections. For as the example of M. Justin Trudeau amply demonstrates, no PM can be trusted to make good choices. ..In a country with two official languages, he insultingly chose as Gov~Gen someone who only knew one. (It should be noted there is no formal means of deselection — the appointed ones serve out their term, unless death intervenes or they can be persuaded to resign.)
One residual power that does arise sometimes is the granting of a request from the Prime Minister (or Premier of a Province) for Parliament to be suspended for a time, or prorogued as it is called. This is not a rubber stamp situation, for the request may be to avoid a vote in the legislative body that the government is likely to lose. So whether or not to grant a prorogation depends on the specific situation.
In December 2008, then PM Stephen Harper asked such a thing of Governor-General Jean. Arguably she should should have refused, since the Liberals and New Democrats had signed a formal agreement to form a coalition government in the event the government fell (as it well might have, being a minority government). …But the situation was clouded due to uncertainty regarding the Liberal Party, for it was the outgoing leader who had made the agreement, one the new leader (not yet chosen) might not be committed to. A recess during which the situation would become more clear did make sense. Granting the request for prorogation was the sensible choice then. But it might not be another time, in other circumstances.
So yes, every so often the Crown’s representative does have a momentous decision to make. And no, I would not want to see an elected President deciding the fate of an elected Prime Minister. The monarchical form of government avoids that sort of awkwardness.