As for the "but Cullen is pro-Templar" that's ... not necessarily the same as pro-Circle. He also, with a mage Warden, was once open-minded enough to be in love with -- or at least infatuated with -- a mage. And finally, he's the slightly awkward, emotionally vulnerable, literal knight in shining armor -- with the bonus tragic bits that you can "fix." Of course he's popular.
lol
Disclaimer: I'm currently doing the wangsty, leave-him-on-lyrium romance. So I don't hate him either. But, c'mon. Sharp-tongued ambitious lady who is not a LI, or broken-yet-fixable knight LI ... hardly a fair comparison. 
I didn't romance Cullen, but I liked his character in Inquisition. He's reasonable, he's trustworthy, and he's able to accept that the protagonist has different views about mages and magic than he does. Unlike Vivienne, Cullen doesn't act like a child simply because he doesn't get his way (for example, when the Inquisitor sides allies with the mages instead of the templars).
Despite the fact that I'm pro-mage, he was someone I trusted. There's a stark difference between Cullen and Vivienne, and I simply don't think one can say that people only dislike Vivienne because she is pro-Circle (although I think it's more fair to say that Vivienne is pro-Vivienne, and simply uses the Circle to help promote herself).
Iron Bull also explains to you how mages work in Dalish clans, lol. And ... Vivienne's not wrong.
No, Iron Bull explained how it worked with the mercenary Dalish's clan, not with all clans. When Minaeve says that her clan exiled her, the elven Inquisitor can tell her that his didn't operate that way, and in the dialogue with Vivienne, the main character can make it clear other clans operated the same way as Clan Lavellan as well - which is evident from Merrill's own codex entry and the example of Merrill's original tribe, Clan Alerion. As it reads, "As each generation passes, magic becomes more rare among the Dalish. As the gift dies out, talented children are moved between clans so that every Keeper has a successor, and no clan is in danger of being left without guidance."
However, when the Dalish Inquisitor points this out, Vivienne thinks her hearsay trumps his actual first-hand knowledge of the People. She's wrong because she thinks she's right simply because she's heard rumors about the Dalish, even when she's presented with actual facts about the Elvhen from one of their own.
I do like her as a character. I probably wouldn't have much in common with her at all IRL, as I find worrying about what people might think of X action or X outfit or whatever exhausting and shallow. I think I'd still appreciate her verbal jabs, because I've actually been zinged really good during RL arguments and had that end the argument because I laughed so hard. I appreciate that nonsense.
Vivienne isn't without either flaws or merits. I don't think she's quite deserving of some of the level of vitriol some feel toward her.
I'll be honest, I think she's earned all the disdain and vitriol that is aimed at her. People dislike Vivienne for who she is; it's as simple as that. It doesn't help that her writer thought that throwing tantrums and making snide remarks when the protagonist doesn't agree with her somehow equated her to being a master of the Game, when it actually does the opposite.