His problem isn't that they have a culture - is that they think their culture is his culture, and that he is wrong for suggesting otherwise.
Problems start when Dalish believe themselves to know more than anyone else, hold their view is the right one and use it to decide what is really Elvhen and what not ("Never mistake them for arbiters of "true" Elvhen culture").
I'm sorry guys, but I've never seen a Dalish character claim that their culture is a pure, perfect, complete representation of "true Elvhen culture," nor that they think they've already got all the answers to the past and don't need to learn more. If you have seen a Dalish character or two who thinks that (maybe from a novel or comic), then I'm sorry, but they're the exception.
Everything I've ever seen and heard about the Dalish, from codex's to character testimony, is that they know they don't have most of the answers to the past. They know their culture is an incomplete reconstruction of ancient elven culture. They want to learn and discover more about their past (and are constantly digging and searching) because they consider their culture a Work In Progress in reconstructing the Dales and Arlathan--yet everyone accuses them of claiming they already have all the answers and they're a complete reconstruction of Elvhenan?
At worst, they're condescending because they believe their culture knows more about ancient elven culture compared to others. And, yeah. Before ancient elves made themselves known, they had every reason to think so. Most dwarves and humans and Qunari don't care about anything "elfy," so of course they don't know as much about ancient elven stuff. Most city elves want to retain their heritage too, but it's established in-universe that most city elves have forgotten most of their history and lore.
However, they're still open to admitting they were mistaken about the past, like bringing the letter from the Knight's Tomb to the Dalish, who immediately accept it as a relic of their lost past and send an apology gift to Red Crossing (after centuries of claiming that the Chantry's claim of an elven attack on Red Crossing being propoganda). The Trespasser end slides also show that that both city elves and Dalish elves flock to Fen'Harel's side, rather than all Dalish rejecting him because "we've already recreated the past / we don't need you / Fen'Harel is the villain of our lore, so we're not trusting you."
I also don't really find Solas to be a reliable narrator for his "I tried to share knowledge and they cruelly spurned me" tale of woe. Much as I love him, he conceals the fact that he's an ancient elf, which doesn't really establish good ethos with people you're trying to win over. Also, from what we've seen in the game, he can come across as a pretty arrogant and condescending know-it-all himself.
He doesn't blame them for getting it wrong, given the many horrible things that happened to the elves in the past. In fact, his dialogue provides several occassions in which he commends them for trying ("The Dalish remember fragments of fragments, but that is more than most"). A Dalish Inquisitor romancing Solas can argue that Vallaslin now mean something different for them, and Solas concedes the point. That's not his issue.
Yes he does. When you first talk to him about elven culture, he'll express disdain for the Dalish, saying things like, "while they're busy picking through ruins, I've gone into the Fade and experienced the past firsthand," and called them "children acting out traditions they don't understand." I was surprised that there was no option in-game to point out it's ridiculous to hold it against the Dalish as a culture for not Fade-walking the way he does since it's common knowledge
He does concede later that they're trying, but that's further along in the game, usually after a Dalish Inquisitor befriends/romances him.