I agree that the characters are important not just for what they said, but what they did. Although having generic, inter-changeable 'talking head' dialogue, which 100% of Jadzia's dialogue was, is always a sign of bad writing/bad character and universally makes for bad entertainment regardless of if her actions had some merit of her being a character that exists on the show.
However, Jadzia fails on the "what she did" front too. She never did anything interesting or important. She did nothing that couldn't have been done by any of a whole slew of other characters. If O'Brien had 100% of her lines, they would have had to make very, very few changes to the scripts and how the scenes that they shot were shot, or what they shot. Anything that couldn't be given to O'Brien (ie: 'friend who listens to Sisko') could certainly have been given to Kira. There is no other main character on DS9 who could have been so easily removed (other than Ezri); because all the others are way more useful & intergral to the plot of the show.
When you throw around numbers like 100% you put yourself at a disadvantage in this debate; you charge those who seek to disprove you with the easy task of finding even just one example (you could look
here for some examples. It's not just that these lines come from a science officer or from a Trill, but that they come from Dax). O'Brien wouldn't have been the "friend who listens to Sisko" because they needed a character with a long history with him, that's one of the main reasons Dax was on board. And even if they could just switch these lines to someone else, why would they? They had a character that served a role.
It must be a very personal distaste for the character for you, and that's fine. But to suggest that she's useless and does nothing to further the show is simply incorrect.
It was correctly stated above that she was really the only person that could serve as a sounding board during "In the Pale Moonlight." The only three among the crew that could do it would maybe be Worf, Kira, or O'Brien. All three of them have demons of prejudice that they would have had to deal with. Dax was much more objective and insightful into Romulan philosophy and historical context, not to mention Sisko's closest friend. It couldn't be Garak or Quark because he hadn't considered going outside of Starfleet at that point yet.
To determine whether or not a character is useful is to ask if they have a point of view, something that motivates them, and if that motivation complements the show through that character's interactions.
Dax, much more than any character, served as a voice of calm reason having lived far longer than any other character, and long since having grown used to her own skin. She has not become bored with life lamenting her long life like Quinn the Q. Instead she embraced life and seized every day, learning to live with moderation without getting bored, and empowered by her lifetimes of accumulated knowledge and natural intelligence.
But the OP was referring to her role as science officer. The above suggests that she was very strong character, job title notwithstanding. You argue that any of her lines could be uttered by O'Brien or Nog. Maybe while on the bridge (though even on the bridge they managed to get some character based dialogue in, believably demonstrating a crew waiting for the storm to come), but then who really cares? They are after all trained Starfleet officers. In a real pinch all of their lines could be interchangeable to some extent. One doesn't watch a show just for the bridge or ops scenes or there wouldn't be any reason to do 178 episodes. That Dax served a purpose long after science became a secondary priority only boosts her as a character, both professionally and in terms of her relationships with the rest of the characters.
Some examples of quotes from IMDB that reveal to me the character of Jadzia Dax (most of which couldn't be spoken by Miles Edward O'Brien with all respect due to him):
Major Kira: [
after revealing that she and Shakaar have split up] The way I see it, people are either meant to be together, or they're not.
Lt. Commander Jadzia Dax: I guess I'd rather believe that any relationship can work as long as both people really want it to.
Lt. Commander Jadzia Dax: If I were in your shoes, I would be looking for someone a little more entertaining, a little more fun and maybe even a little more attainable.
Lt. Commander Worf: You are not in my shoes.
Lt. Commander Jadzia Dax: Too bad. You'd be amazed what I can do with a pair of size 18 boots.
Sisko: All right, say it.
Lt. Commander Jadzia Dax: What?
Sisko: That I have lost all my perspective, that I'm turning this into a vendetta between me and Eddington and that I am putting the ship, the crew and my entire career at risk, and if I had any brains at all I'd go back to my office, sit down and read Odo's crime reports.
Lt. Commander Jadzia Dax: I wasn't going to say that.
Sisko: But that's what you were thinking, right?
Lt. Commander Jadzia Dax: No. Actually, what I was thinking is - you're becoming more like Curzon all the time.
Sisko: I don't know how to take that.
Lt. Commander Jadzia Dax: Consider it a compliment. And the next time I go off, half-cocked on some wild-eyed adventure, think back to this moment, and be a little more understanding.
Lt. Commander Jadzia Dax: Maybe it was arrogant to think you could find a cure, but it's even more arrogant to say there is no cure just because you couldn't find one.
[
Dax is trying to work, but Virak'kara, a Jem'Hadar, is staring at her]
Jadzia Dax: Am I really that interesting? You've been standing there staring at me for the last two hours.
Virak'kara: You are part of my combat team. I must learn to understand your behaviour - anticipate your actions.
Jadzia Dax: There must be something you'd rather do. Maybe get some sleep?
Virak'kara: We don't sleep.
Jadzia Dax: How about getting something to eat?
Virak'kara: The white is the only thing we need.
Jadzia Dax: [
pondering] Don't sleep... don't eat... What do you do for relaxation?
Virak'kara: Relaxation would only make us weak.
Jadzia Dax: [
aghast] You people are no fun at all - I'm glad I'm not a Jem'Hadar woman.
Virak'kara: There are no Jem'Hadar women.
Jadzia Dax: So what do you do? Lay eggs?
Virak'kara: [
proudly] Jem'Hadar are bred in birthing chambers. We are able to fight within three days of our emergence.
Jadzia Dax: Lucky you. So let me get this straight: no sleep - no food - no women. No wonder you're so angry. After thirty or forty years of that I'd be angry too.
Virak'kara: No Jem'Hadar has ever lived thirty years.
Jadzia Dax: [
puzzeled] How old are you?
Virak'kara: I am eight.
Jadzia Dax: I would have guessed at least fifteen.
Virak'kara: Few Jem'Hadar live that long. If we reach twenty, we're considered honored elders.
[
leans in]
Virak'kara: How old are you?
Jadzia Dax: I stopped counting at three hundred.
Virak'kara: [
amazed] You don't look it.
Jadzia Dax: Thank you.