I can understand that side but I think Dany is pulled up on her mistakes. Her council often make her aware of the wrong decisions she has made, decisions that other characters (particularly Jon Snow) make too. I found this and it explains my point of view perfectly -
Jon, in ADWD, is in a very similar situation as Dany and actually makes many of the mistakes Dany makes. He ignores the social and cultural differences between the Wildlings and everyone else (similar to how Dany ignores the differences between slaves and freedmen).
He ignores the council of his stewards and shuts them down even though he knows they resent him for it, and he makes no attempt to cull this.
This is similar to how Dany shut down Barristan for a brief moment because she didn't want to hear him speaking good about Ned Stark... you know, the guy who led a rebellion that slaughtered her immediate family and forced her and her brother to live in complete fear for her entire life, in abject poverty and constantly moving from one place to another in order to survive?
They both however, do what they do out of noble reasons. Dany wants to give the slaves a chance at freedom and to give them an equal opportunity to earn a living by using their skills to carve themselves a position in society and break the social divisions that bind them.
Jon wants to save the Wildlings from The Others because he considers them, above everything else, men and his duty is to guard men from The Others. He doesn't want to sentence thousands of men, women and children to die a horrible death.
And yet, the criticisms of Jon's actions never go so far as they do for Dany, even though their story arcs in ADWD are almost the same. Sitting around solving political problems and turmols after a difficult struggle.
Though I have taken this from online, I think it speaks truth about the criticism of Dany. In many ways, I feel that fans of the books and the shows want to dislike her because of the power she has and the "birth right" (depending on opinion) that she bears.
Additionally - correct me if I'm wrong - it can be said for some fans that Jon Snow is favoured because of his status as a "bastard", as an underdog in the world of game of thrones. For me, Jon's life is pretty comfortable compared to that of many who live in such a world and he definitely shouldn't be seen as an underdog. Sure, he's had his struggles and battles but so has every character, so why be so supportive of him, especially when his bravery, triumphs and victories can be matched by other characters.
In reponse to you saying that Dany is the one we are meant to be rooting for or a Mary Sue as you say, I disagree. I believe that Jon is the one we're "meant" to be rooting for. I think his flaws are not as obvious if apparent, at all, compared to Dany's and are certainly not discussed as widely. Jon is a poster boy (or Marty Stu, which ever you prefer), he always tries his best, always succeeds and has climbed the "ladder"; he's brave, honourable and doesn't like killing but "has to" for (what he thinks) is for the good of the people or because it's the right thing to do. Dany, pretty much mimics this throughout the series. So, why is it that Jon is given a free pass when it comes to his actions?
At the end of it all, they are both strong characters and I suppose my main issue here is that I don't see why fans of the series can criticise Dany for the mistakes that she has made, when another one of the fans favourites carries out similar sorts of actions which result in similar sorts of consequences.