If the Prime Minister of England or the President of the United States came out and publicly said "don't say such things about me, you're totally wrong," would that change the opinions of the people about him or her?
It does not. Rather, it would probably be taken as either a new point of criticism or a statement of humility, depending on whether the person love the president or hates them.
Political leaders say things all the time that are either taken out of context or are ignored by the vast majority of people, or is not taken at face value.
It didn't bother me that my agnostic Qunari who kept saying he wasn't Andraste's Herald was ignored on that front because he had become a symbol in the hearts and minds of thousands of people, and as the scene with Iron Bull dressing my Inquisitor up as a mercenary and going to drink with the grunts, and the Inquisitor remaining utterly unrecognized, shows that the Inquisitor is a symbol or an idea more than anything else in the eyes of most.
Ok, now imagine a soldier telling a general, "I believe you are wrong, sir."
There are things you can believe and then there are things you can say. Contradicting your leader right to their face...
If the entire Inquisition doesn't listen to its leader, if they continue to talk about it in from of their leader... then that only shows a severe lack of respect and that is something that should be addressed by someone. But no one does. Instead you get condescending comments like, "Let them believe it, it's good for morale." as if you didn't even matter. Be a good figurehead, do as you're told and leave the adults to running things.
No one asks for your permission, your opinion or anything. Oh, they listen to you and then go and do what they want anyway. Remember when they asked you to be the Inquisitor? No? Yeah, because they never did. They decided it and you didn't get a chance to say otherwise. Freedom of choice, my ass.
Remember all the nobles visiting Skyhold? Remember getting advice in dealing with them from Josephine? How often was that useful? Never? Yeah, because they never actually let you meet anyone.
Of course, no game can offer all possible choices. But at least it should leave you the illusion of choice and even here DA:I fails completely. Oh sure, you get to make a few decisions (mage or templar) but in the end none of them mattered.
The only way to make it through the game with a goal in mind requires meta gaming and/or knowledge of the novels. For someone who has access to advisors and a spy network you're lacking intelligence badly. "You have talked five minutes with each candidate... who's going to rule Orlais now?"
Meh.