I think it's more than that. As Carmack observes, you don't
see in 3D. You see two 2D planes that your brain extracts a certain amount of depth information from. As a UI goes, you can't actually freely use that third dimension, because as soon as one element obscures another, either the front element is too opaque to see through, in which case the second might as well not be there, or the opacity is not 100% in which case it just gets confusing fast.
You've really got something more like 2.1d to work with with humans; intefaces need to take that into account, and maybe sometimes use that .1d depth dimension to provide light hinting, but not depend on it.
So you're not removing a dimension... you're acknowledging it doesn't exist. There isn't really a "third dimension" here to take advantage of. It may be visually impressive if something flies from far away to up close, but in interface terms, all that has happened is that the z-buffer values shifted, but the interface still has the same number of square radians available that it did before.
The depth information is a lot more like an additional color than an additional spatial dimension. UIs are better with color hinting available when used well, but there's not much fundamental difference between a color 2D interface and a monochrome 2D interface. Proof that it isn't much is that there are some people who use settings to set their phone to monochrome, and the result is at worst occasionally inconvenient, not crippling. Compare with trying to go from a 2D interface to a 1D interface, which would be utterly crippling and destroy a huge swathe of current UI paradigms.
To truly "see in 3D" would require a fourth-dimension perspective. A 4D person could use a 3D display arbitrarily, because they can freely see the entire 3D space, including seeing things inside opaque spheres, etc, just like we can look at a 2D display and see the inside of circles and boxes freely. They have access to all sorts of paradigms we don't, just like no 2D designer would ever come up with a "text box", even though it fits on the 2D plane, because no 2D user could see inside the text box to see what's in it.
(This is one of those cases where the mere act of typing it out improved my understanding of the topic. I didn't have the "depth information is really just another color" idea going in, but the more I think about it the more sense it makes, in terms of the amount of information that channel can carry. Just like color, it's not quite 0, but you can't really stuff all that much in there.)