Look at most of the asshole leaders and tyrants in the world, and they are usually the presidents of republics. Fuck republics !
But I'll say this - the most visible part's of the Governor General's job are actually the least important.
Most of what people see is about ceremony and various expressions about Canada's tradition. Presenting Orders of Canada, having ties to the Queen, etc. - this is cultural stuff, but it is not the deeper reason.
The most important is the stuff that never or almost never happens:
There's constitutional role [usually after elections], as PM-wannabes try to form a government (constitutionally, meaning a cabinet with themselves as the leader). Most times, the choice is obvious - but in a minority situation, or when a government falls due to losing "confidence" then a GG or LG is faced with the choice of some new configuration or else calling a new election. The party leaders can ask, but that does not always mean they get. Just look at how Christy Clark's time ended as Premier.
That's the second-most important part. The first most important (and not yet ever used in Canada, thank fuck) is what 80Watt mentioned - the fail-safe against tyranny by a PM.
The Governor General is the actual Commander-In-Chief of Canada's military. Above the PM and Minister of Defence, who may de facto give the orders, but are not at all who the armed forces swear an oath to.
If a PM ever tried to pull something that was an absolute violation of the constitution, a betrayal of the nation, or a grab for pure tyranny, they would need to have the military on board, but the military is not obligated to carry out any order like that.
A Prime Minister is just one more member of cabinet, and can be removed by losing confidence of Parliament at any time. But what if they will not go, or try to usurp Parliament? Well, there is still that one last "fire alarm" that the GG (or the Queen herself) can pull. An order to remove the PM, or just to not obey their orders. If that happens, it is game over. I mean, this is true world-wide of any government - if you lose command of the military you lose power, period. But soldiering is one of those things that depends very much on oaths sworn to the chain of command, so if your actual commander is actually above the wannabe-tyrant, there is no disloyalty to your country if you disobey the fucker. (By contrast, look how close to that level of crisis the US just got. Their oath is to the constitution but the President personifies their state.)
Unfortunately, I think we do have a flaw having a GG and not the actual monarch, in that it falls to the PMs choose reccommend the GG to the monarch. This means that although the tradition is to to choose some kind of eminent non-partisan person, the choice also leans very much on optics (and that's an old tradition too).
Julie Payette was chosen because 1) the [unofficial] tradition was that they alternate between anglophones and francophones, and man and woman, and 2) she was a fucking astronaut, which is about as eminent as you can get. Should they have vetted her better, to detect those "issues" in her past, and predict a possible bad future? Sure. But think about what kind of vetting it takes to become an astronaut in the first place? Massive background checks, massive psychological examinations, a person has to be a team player, and ethically above reproach. I guessing this is what they told themselves instead of vetting her with some sort of deep dive - that NASA standards were so far above their own there's no way the choice could be wrong.
And then, when the bad news began to flow about Payette, they probably told themselves this was just the Conservatives' (media included) usual pissing and moaning about anyone who is francophone or female occupying the GG chair (as they did for Adrienne Clarkson and Michaelle Jean). Well nope, turns out Julie Payette had some very punch skeletons in her closet - perhaps not the usual Ottawa elite sort, but rather the kind you get with an astronaut's hard-driven personality. Not diva behavior, but rather the kind of "ceremonies are stupid, I command you obey, tears are for those who wash out of training" kind of personality. A GG needs some social finesse and diplomatic skill, and instead they hired-blind someone who's as delicate as an anvil.
It's not a terrible mistake, and I don't think it damages the GG office like people say, but they only had to scratch the surface a little bit to have avoided this. Optics only work out if you take a close look yourself.