The Inquisition did save the day, but it only needed to be saved because it continued to exist. In the meantime, before any serious threat to Southern Thedas had been acknowledged by anyone, including the Inquisition, you keep the Qunari body a secret. It might be fine if it was in some dingy wooded area in the Dales, but there was a dead Qunari in the middle of Halamshiral, the second most important city in one of the three most important nations in Thedas, who was found in during the most important political congregation in the South since the peace talks two years ago.
{OK, suppose you do tell the council about the Qunari from the get-go. Why, pray tell, is it up to you to sort it?}
Then, once you establish a threat, the gaatlok plot, you keep it a secret and your forces take over part of the Winter Palace by force (and again, 1000% sovereign Orlesian territory) without actually explaining anything.
And those are the things that must happen. You have choices that make it even worse, like arresting the elf who was handling the gaatlok, as if you had any authority to act in this situation. Why would you (and again, in Halamshiral)? You would because it's linked with this Qunari thing only you know about. You continue to keep it hush-hush and so it appears the inquisition has randomly arrested some elf and overridden the Orlesian guards, which does absolutely nothing to allay the fears of those who believe the Inquisition is over-stepping it's mandate - which, I will remind you, was fulfilled two years ago. A second choice, choosing not to tell the council about the Qunari plot to blow up the palaces of Southern Thedas. A third, to not tell them the Qunari are targeting you directly. They find out, and they're rightly furious. The fact that you continue to exist as a person and an organisation has culminated in a Qunari attack on Halamshiral, and a means to conquer Thedas.