I used to like Varric quite a lot, he was my #1 favourite character when DA2 was all the hype until Inquisition came out and 1) he showed prejudice against Dorian for being upper class and 2) he was constantly trying to prove Cassandra was a naive woman, unable to have real friends.
I wonder what he considers himself to be, then, befriending thugs, smugglers and pirates, like Isabela, who abandon their friends when things get dire.
In DA2, if you manage to have rivalry with him, you find out he's a gossiper who tries hard to ruin your reputation in town. That's a sucker punch from someone who claims to be just your regular surface dwarf, taking care of his own business.
Varric is duplicitous and clearly has a problem with people who come from different walks of life than his own. For someone who's supposedly sociable and chatty, who doesn't care who you are or where you're from so long as your words correspond to your actions - he proposes a partnership to Hawke, even if he doesn't know much about his background -, he's very hostile if you happen to have a noble ancestor somewhere in your bloodline. A funny thing, since his parents are considered to be a wealthy and succesful merchant house. Double standards?
He's not humble at all, nor does he judge everyone by their character, as he likes to demonstrate. Cassandra was more forgiving and honest in her evaluation of him than he was of her. She says she wants to change the Chantry so that Andrastians like Varric can join it someday. That's an admirable attitude for someone who demonstrates contempt for his dishonest ways. In return, what does Varric have to say about Cassandra? A zealot who has no friends. Sorry, but that's as blind as someone can be to a person's true character. And the blindness is voluntary.
I guess everyone has an ugly side. It's just a matter of finding out which characters' faults you can handle better and which ones you simply can't tolerate.
I agree that his anger at Cassandra in DAI got tired after a while, but it was understandable. She unfairly imprisoned him until he finished telling his story, she was fairly threatening during much of the interrogation, even though she didn't physically harm him. And she's asking him to rat out his best friend. After Bartrand's betrayal, Varric isn't going to betray those close to him.
I have never played the rivalry path for Varric so I can't say how negative his actions are. The game does make it very hard to run a rival relationship with him, though. Rather than just being pro-mage or pro-templar or anti-slavery like most of the other companions' influence changes, Varric's influence is more specific to each situation and it takes a lot more trial and error (or metagaming) to select rival actions. I
I didn't see him as a dwarf just minding his own business affairs in Kirkwall. It's fairly obvious fairly quickly how many pies he has his fingers in. He has done work for the city guard, has had agents keeping Merrill safe on her way home, helps Anders' clinic. I think he prefers to work behind the scenes and not get involved in direct politics, which is why he's reluctant to become viscount in Trespasser and why he doesn't like going to dwarven mechant guild meetings. But he does like the intrigue and influence from working behind the scenes.
You seem to be contradicting yourself as you say he is intolerant of people with backgrounds different to him, but then say he befriends a pirate like Isabela, clearly from a different walk of life than him. And he knows full well Hawke's noble lineage when he meets him/her for the first time. Gamlen Amell was still living in Kirkwall when he went broke and it was a well-known scandal of Leandra eloping with a mage. This was one generation ago, it's still fresh in the minds of the nobility, including the well-connected Varric. He has no issue with Hawke being from a noble family. And by the time he meets Hawke face to face, Hawke has been working with the mercenary group or smugglers for a year and has gained a reputation for being good at what they do. He has been looking into financial backers for his brother's expedition, so he's done his homework on the potential funding Hawke can bring in.
I didn't feel him necessary as a companion in DAI, but I view his change in personality as the shift in the PC's relationship and perspective with Varric.
In DA2 we were buddies with Varric. In DAI, he meets us as a fellow prisoner conscripted into a Chantry-affiliated organization far from home. But we are soon elevated to a religious figure and become the face of the organization who conscripted him. Granted, he is free to go at any time and stays because he feels at fault for releasing Cory and red lyrium, but that doesn't mean he can't still be resentful of the Inquisition.
We're not good friends with Varric in DAI. I think he will always keep a barrier between himself and the Inquisitor, as he says he sees us more of an icon than an actual person. The books he writes about Hawke were pulp fiction crime novels, the everyman's struggles in the world. The books he writes about the Inquisition are sweeping epics with larger than life characters.