(@eyebulb, eyy, web stuff’s a bit on the back-burner for me lately, so it takes me a while to get to things, also, heads up, a lot of links)
Well, I mean, how to put this… The symbolism is the point? For me at least.
Like, the whole reason why Zutara is so loved, so enduring, and people were—and still are—so upset over canon is because the way the narrative is constructed paints them as perfect for each other:
You have the not-at-all subtle foreshadowing with the story of “The Cave of the Two Lovers”, that introduces a mythical/mythological aspect into their relationship, which in a story as full of mythical, mythological and spiritual themes as AtLA causes it to make far more sense than any other romantic relationship for these characters on a meta-textual level.
Not only that but by making it a reincarnation romance that has already ended in tragedy once, having them get their happily-ever-after now ties neatly into the general themes of moving forward and new generations undoing the hurt of the old ones, and does so in a beautifully poetic manner.
Speaking of tying in neatly into the themes of the show, a relationship between Zuko and Katara would be a very beautiful, very poetic, very symbolic illustration of the “Illusion of Separation”. Mainly, Zuko and Katara have, continuously throughout the show, been learning from one another* and coming out better, stronger because of it. It’s only natural to assume that this would continue, and in fact get only more pronounced as they grow closer.
*Most obvious example being all the waterbending moves Zuko uses in the Book 2 finale.
Then you have the way their stories mirror one another—their bending*, their deep burning sense of justice, their mothers (more on this later), and the way they both hinged all of their hopes and dreams on the Avatar.
That last one is especially interesting because it sets them up, for the first two seasons (a.k.a. the Half-Of-The-Best-Series-Ever-That-Got-Followed-By-A-Mediocre-Self-Insert-Fic seasons), as mirroring, opposing forces with Aang as the axis around which they revolve; The push and pull, if you will.
*By this I mostly mean how they both took a while to come into their own, how they did it at around the same time, but did it in contrasting ways. Mainly, Katara finally found a Waterbending Master and received a formal education after years of struggling trying to learn from alternative sources, while Zuko after struggling for years with a Firebending Master finally found his footing once he turned to alternate sources of wisdom.**
**This is a stray observation, it’s not all that relevant, but I still find it interesting.
Then, to get to your question, you have Zuko and Katara as individuals.
Firstly, Zuko and Katara are, fundamentally very similar.
This is a good thing.
They are driven, determined, have a strong sense of justice, and above all, are deeply emphatic and compassionate people.
These are important characteristics to share with a significant other. Having similar goals and aspirations is a fundamental part of a good and stable relationship.
(More on their similarities here, on Zuko’s character and its misinterpretation here and here and on Katara’s character and its misinterpretation here. Also this post, just so we’re all on the same page.)
Secondly, they have experienced a similar trauma. This, combined with the above mentioned empathy and compassion, helps them reach a level of understanding that they do not have with any of the other characters in the show. Least of all, ironically, their canon love interests. (More on this here, here, here, here aaand here)
Finally, there is the fact that the very plot of the show keeps pushing them together:
They spend every season finale fighting together (in all meanings of the phrase), their most emotionally charged moments are spent together, they are the characters that are shown having the most equal, mutually supportive relationship of all and, above all else, the actual, real emotional climax of the entire show is hinged on their relationship.
So, when you take all of these elements, you get one of the most beautiful, meaningful and complex romances in animated television… that was never made canon because the creators’ petty self-insert fantasy overrode any scrap of creative integrity they may have had.
Oh, well. It ensured that the Zutara fandom to this day remains one of the most enduring, productive, creative fandoms in existence, so the loss doesn’t sting as much as it could’ve.
Well, hopefully this helped you to at least somewhat understand where Zutara shippers are coming from, if you’re interested in some additional reading, I can highly recommend @theadamantdaughter and @araeph‘s zutara (and AtLA in general) meta, as well as my Avatar meta tag, or feel free to hmu, though as you can see, it may take me a while to answer.