I can see what you mean I suppose and while the vast majority of her moveset could be traced back to the games I wouldn’t call any of them as integral to her identity. Even the parasol has stopped being her signature move.
but I would also argue that nearly the entire roster has moves that aren’t integral to their identity
Yup, that's true in a way. But Zelda feels like she ONLY has moves that aren't integral to her identity. Or rather, moves that honestly feel like... chosen just because she needed to be in Smash. I mean, sure, she's a mage. Cool. At least we don't have that many.
Generally, when characters have many moves that aren't integral to their identity, I put the blame on the philosophy Sakurai had while developing older titles. I can clearly see a shift in this sense from Brawl onwards. However with Zelda, the blame is all on Zelda herself and her role in the series. There is no way you could make Zelda work in Smash without picking generic references and bland spells just to make her work. Arguably, the only move you could use for her is the Light Arrow as a neutral special. But that's it.
For characters like DK, Samus, Ganondorf, or even Ridley in a way you could definitely add change some stuff to make them feel faithful to their source material.
But please, do note that faithful to their source materia =\= having their moveset filled with references. For example, a character that feels very close to their source material is, in my opinion, Joker. In Persona 5, Joker and the game as a whole are all about style, cockiness in a way. And Joker (while he could use his main attacking animation as a forward Smash) embodies so in Smash too. His moves are stylish, flashy, flow along very well. Even Arsène as a gimmick makes sense, despite not being faithful to his home series.
Going back, Marth uses a few clever references (Shield Breaker may reference the Rapier's effectiveness against armored foes, Counter wants to replicate the hit-counterattack flowchart of FE, though with debatable results) that make him somewhat feel like Marth in his source games. However, Marth feels right. He is a vanilla swordsman because his role in the first and third game of the series is just that: the righteous leader of the army that happens to be a swordsman. Even his stats are balanced!
But Zelda? What is Zelda supposed to feel like?
My issue with Zelda in Smash isn't all that related to Smash. It's related to Zelda's role in her home series as a whole. I probably will never be able to properly express my thought on Zelda, but it's a mixture of reasons, which can be boiled down to:
1. Zelda has a marginal role in her home series (she's mostly the namesake of the series, but in the games she does very few things before the endgame)
2. Her special moves (with the exception of the phantom, which is a good adaption of the move) are taken straight from Link because she lacks solid references.
3. Her standard moves are bland and uninspired magic attacks because Zelda is a spellcaster, though this is a rather minor part of her appearances. This is especially true considering that when Melee released, we only had gotten the first Zelda game, Zelda II, A Link to the Past, Link's Awakening (in which Zelda's absent), Ocarina of Time, and Majora's Mask (in which Zelda only makes a cameo), not exactly games where Zelda's spell casting abilities are shown in a significant way. I bring up this point because Zelda's moveset is mostly unchanged from Melee.