I’m not sure I’d be as particular about who does it, Studio Ghibli is probably fine as any other, but I do know Illumination just would not be a good fit. Good enough for Mario, but I just don’t see any facet of Zelda in the same light as your typical extremely lighthearted cg movie with goofy characters and animations and constant action or jokes. Even Wind Waker wouldn’t quite fit that.
That being said, others still seem to expect a LotR vibe. Not quite sure I understand that, either. The only time the series has really gone for that has been Twilight Princess and I honestly found it very boring and not really staying true to that aesthetic in overall tone anyway. Zelda is not super grim, dark, ultra epic or entrenched in generic vanilla fantasy either.
Breath of the Wild/Tears of the Kingdom I feel has struck a pretty good balance in my opinion. It’s whimsical, melancholy, playful and adventurous. Of course, I feel adapting any one particular game is just going to restrict them. I find it especially funny seeing others saying, on other forums, “they’d have to split it into three movies to get all the dungeons in! It won’t work!” - ideas like that, people are just not thinking about how films actually work and I’m so glad they are not in charge. When talks of Shadow of the Colossus were going on, the fact that there are so many colossi is a bit more intrinsic to its narrative, and otherwise it’s a pretty quiet experience, so that would certainly never work, but Zelda is not quite the same.
The Mario movie had its own story, basic as it was. I prefer the two Sonic movies, and although they loosely adapted some bits from the early games, they are still largely their own stories, and I’d say definitely better than anything from the actual series for the past couple decades. If the same or similar ideas could be applied to a Zelda movie, I have no doubt it could be just fine. Don’t go changing any set elements too much, but don’t let the lore restrict everything either, find out what the core of the story is about and let that dictate the pacing.