All three choices are exactly the same, we get a few extra (unnecessary) soldiers no matter who rules. I realize the excuse used for allowing us to fail our goal of stopping the assassination is that Gaspard will be blamed and that somehow that will make a huge difference. I just think it's a very weak excuse. They're already at war, we're told there is already assassination and such happening behind the scenes in "the game" so why would it matter if Gaspard was falsely accused rather than him winning the war and having her executed or him having her assassinated for real as part of the game? Why would any of it matter to the inquisition? The whole reason that bad future happened in the time traveling quest was because the inquisitor "died" so there was no one to seal the breach or kill Corypheus(because reasons?). The not-dead inquisitor curb-stomps Corypheus and seals the breach for good before anything major in Orlais would have happened and it wouldn't hinder the inquisition's goals. So we're not given a strong story related reason to be doing this quest, we're not given a strong personal interest or connection to this quest, so IMO this quest is useless. The plot of DA:I for the most part felt like a mishmash of random unconnected events with only a half-hearted excuse to do any of them and no personal stakes.
It's not a weak excuse. It doesn't even require a lot of explanation. It's political legitimacy. This is the single most important currency in politics, and we see it play out every day in the media, whatever country you're in.
It matters a great deal if people think Gaspard did it, versus if they believe he did not. It's the difference between him continuing to have an open rebellion on his hands - making his assistance totally useless - and just having bitter malcontents who fought on the losing side of a war to deal with on his return. Even without the demon army, the balance of power in Orlais shifts a great deal depending on how legitimate his reign is as a ruler.
And remember, the original plot is for his sister to sell him up a river.
And you forget what happens without the Inquisitor:
1. The breach is never temporarily sealed. That makes things worse all on its own.
2. Corypheus gets a demon army.
3. Orlais is in chaos.
The reason Corypheus can't be killed is super obvious - it's quite literally part of the main plot: you don't know how to do it until the Well of Mythal.
And we're given an explicit reason to do this: the Inquisition does not have the forces to actually fight Corypheus, notwithstanding the fight at Haven. With Orlais in chaos, you don't have the support, ultimately, to match up with Corypheus. Plot-wise, you can't win at the Arbor Wilds, and so Corypheus gets to the Well first.
This is as good an explanation for why you don't need Orlais in DA:O, or why you do need the elves and dwarves and why a second round of purely human Fereldan levies isn't enough: total plot contrivance because this is the story the writers want to tell. You either buy in at the start or you don't.
(sorry, I know I'm not the one you were addressing) I don't know if "personal" is the right word, but there were many times in DA:O that I felt...emotionally invested? In the NPCs and situations where in DA:I I usually didn't feel anything or care about what was happening. I'm sure others will disagree or not see the difference but here are some examples:
I'm just going to say that I disagree entirely. I didn't give a fig about Cailan. He was obviously an unqualified moron, and it's not entirely clear why I should feel bad that he died specifically because of the moronic decisions we saw him make. Depending on your race/origin, his complete incompetence and inability to govern was the entire reason your beloved cousin was raped as a CE. His inability to keep his vassals in line - and Arl Howe basically says this outright - is the reason your family is dead as an HN. Not exactly much to mourn there.
Gaspard is plenty pleasant to you when you meet him too, but that doesn't make it some personal moment.
And let's just say I completely and totally disagree on DA:O doing anything in showing a gritty or dark alive. Remove the brown, and people in Ferelden live incredibly clean, healthy and surprisingly educated lives (seriously, everyone can read?). The only people we see not doing great are the CEs - and they live in the same looking house as everyone else! DA:O totally fails at showing poverty, with the exception of the Casteless.