That moment is legendary among film buffs because it’s one of the most jarring "jump scares" in cinema history. It feels like a horror movie because, in that moment, Xanadu is a tomb.
The "shrieking cockatoo" serves several specific purposes:
The Psychological "Snap": The bird’s scream signals that Kane’s mind has finally snapped. Susan has just left him, and the silence of his giant castle is suddenly broken by this piercing, ugly sound. It’s the audio trigger for the violent room-trashing scene that follows.
The "Eyeless" Gaze: You noticed the eye looks strange—it’s actually transparent. Because of the way it was filmed (a "matte shot"), you can see the background through the bird's eye. This reinforces the theme of hollowness. Like the statues and the castle itself, the bird is a beautiful, expensive object that is dead inside, just like Kane's soul at that moment.
Waking Up the Audience: Orson Welles famously said he put the bird in because the movie was getting too quiet and he wanted to "wake 'em up" before the final act. It’s a bit of showmanship to ensure the audience is paying attention for the Rosebud reveal.
The Exotic Prison: The cockatoo is just another "collectible" trapped in Kane’s zoo. Its scream is the only honest reaction to the suffocating atmosphere of Xanadu—a place where life is preserved but never lived.
It’s the sound of a man-made paradise turning into a nightmare.

1. Since that section of the film is from the point of view of the butler, maybe it's a representation of how he saw Susan, or
   2)It's Orson making sure the audience is awake. I think I read that one somewhere.

The cockatoo's shrill cry is a metaphor for Kane's mind state, which is tantrum-filled (he's going to wreck her bedroom) and child-like. Yes, the missing eye was a mistake by the special effects department.

It's very loud and if you notice, the eye is missing. I think it's a wake up call. It wakes the audience up and warns of an important upcoming event (Susan leaving). I think it represents Susan, the harsh voice and "flew the coop." But I don't think wells meant all that much symbolism in the bird.
On a plain level, it's to wake up the audience. Orson thought he film had become quiet and slow at that point. This is the main reason.
I don't know for sure, but I think the eye was a mistake. I may be wrong, though.