Counter-hypothetical: what if the Monarch decided to act against the Prime Minister and appoint a Governor General to act against their mandate? Both your hypothetical and mine are incidents of "bad-behaviour" going against norms to push agendas. We would prefer were that neither were possible. However in your hypothetical at least the person exhibiting "bad-behaviour" (the Prime Minister) has some mandate given that they were democratically elected. Whereas in my hypothetical the person exhibiting "bad-behaviour" is an inherited position held by someone in lives in a far-away place and may have only set foot in the nation they are meddling in a handful of times.
In either situation we're accepting the risk of bad-faith actors manipulating the structures of power, but if we ditch the Monarch, at least the person doing so is in someway accountable to the people. Harper was successfully able to dodge a confidence vote, but in the end he was ousted from power in a democratic process. I'd argue that's the better scenario.
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.