The issue I have with Leliana and Vivienne is that they completely change core tenets of the Andrastian faith as it is understood in the South. Faith in doctrine and what you have been taught and have come to embrace either through genuine belief or simply be assimiliating it as something you are expected to fall in line with to take your place in and understand society, cannot simply be changed overnight like this. You can't just go to people and say 'Guys guys, I know we always believed in these things here, but now we believe *these* new things instead. Read them over and be ready to start believing them by Mass tomorrow morning okay? Cheers.' And its even worse if you're trying to do so by force!
I agree with what some have said in that Vivienne is the most naturally suited to be Divine. She has intelligence, contacts, faith, charisma, drive and knows how to play to a crowd. The south is reeling from disaster after disaster, Orlais and Ferelden at war, the 5th Blight, the Mage rebellion, the rift in the sky etc etc. They want some peace, but they also want strong leadership in these times. Vivienne would be an inspirational Divine, someone would own the office and make it majestic and important. It would help the South to feel like they were not simply some tramautised, weak nations, weary from war etc.
And in most regards she is not anywhere near as conroversial as Leliana's radical root and branch reforms. But in the one obvious category (i.e she's a mage), she is by far and away the most controversial candidate you could have. Asking for mages to be clerics is bold enough, but starting this out by having a Mage become Divine is one hell of a leap over the line! And how do they explain to the masses, when they have essentially acce[ted the Imperial Chantry's philosophy that Mages can serve from high office, and that Ruling is only really supposed to be intepreted as Tyrannical Rule etc?
For that matter, would the Imperial Chantry not be justified in saying 'Hey, since you seem to have seen the light and accepted the true Chant finally, you don't really need your own Divine do you? Shall we pencil you in for Thursday, so you can all drop in to Minrathous and swear your vows to the one true Divine?'