I've mentioned before that I was somewhat shocked, upon my re-reading of The Lord of the Rings a few years ago, the degree to which Arwen in the text of the novel plays the part of a trophy. Yes, in the Appendicies we get some sense of her "real" story, but in the body of the novel itself, not even slightly.

So, she's not important at the ending of The Lord of the Rings. She might well be* the last named character from the novel to die, but that's not the same thing at all.

*Actually, I bet Tom Bombadil survives her, though perhaps his little bubble of story wanders completely out of Middle Earth before her death. It's unclear what happens to the other immortal magical characters who are not elves--the Ents, Radagast the Brown, and Gwaihir for instance. It's fair to guess that Shelob dies at the hands of Faramir and Eowyn the Slayer, but that's also never stated.