The difference between a bad-actor Prime Minister and a bad-actor Monarch is precisely that the latter, in today's world, clearly has no legitimacy outside of enforcing well-established norms. So a Monarch appointing their unelected friend as Governor General, contrary to the Prime Minister's wishes, would simply result in an extra-legal declaration that the country is now a republic, or possibly that the Monarch is now the next person in line of succession (ie, forced abdication, again, extra-legal). In contrast, a Prime Minister who tells the Monarch to dismiss the Governor General and appoint their friend as Governor General instead, after loosing a confidence vote and refusing to resign, will presumably have the backing of some segment of the population (unless they're just insane), and hence will be much more dangerous, if the Monarch declines to exercise their power to refuse this request.
I do not think it is a good idea to assume that a Monarch will always be viewed as having no legitimacy outside of enforcing established norms. While that is certainly the case now, I would not want to rely on that being true forever. After which we would have to rely on benevolence (or perhaps indifference) of undemocratic executive power.
Could we not solve the problem of the PM appointing a lackey as Governor General with other form of check-and-balance that requires zero input from individuals with no connections to a democratic process? Perhaps a similar way that Supreme Court Justices are appointed (candidates recommended by the Prime Minister and approved by the federal cabinet). While not immune to abuses of power, I would like this better than a Monarch being that check-and-balance.
