Vivienne faces three separate rebellions in the first few months, which is three more than anyone else gets. How is that everyone else being 'cool' with it?
Vivienne isn't a popular divine. What she is is a capable politician who's made allies with those who were willing to support her and examples of those who tried to dethrone her.
But she isn't a popular politician. That's what makes her so great as a character - she's basically an amazing huckster, who's learned to project an incredible amount of power and confidence without having any. She's managed to horse trade ceremonial positions and fight through humiliation and shame to get to where she is - and that's the best she could do in the society she grew up in. I think it's admirable, in a way.
I think that is more plausible because she promises to keep the mages and Templars in line, which is very much what the Chantry wants most at this point. Still a surprising choice, sure, but I don't think it defies belief like Softened Leliana does.
Sure, Vivienne herself being Divine reeks of the Black Chantry, but her actual policies do not. It clearly says the Circles are back in business, just more meritocratic than before. After so much chaos, I could see this appeasing the hardline Chantry elements.
Her first order of business (in the templar ending) is to try and hollow out the templars into, essentially, her puppets. And she represents political legitimacy for the mages. That's something not even Leliana can give mages in their most pro-mage ending. Sure, they're free - but they're a fringe group on the outside of political reality. The anointing of Vivienne returns mages to their 'natural' role as rulers over mundanes.
The Circles aren't back as more meritocratic, because Vivienne doesn't get to where she is on merit. She gets to where she is on pure cunning. What she opens the door to is not capable mages, but political mages, and having the Circle gain true political power. The Chantry fought a war and almost broke apart over that point.