On some lenses the image circle does indeed get bigger as you zoom.

I know from experience the Canon EF-S 10-22 f/3.5-4.5 does this. At 12mm the image circle is big enough for a 1.3x crop APS-H sized sensor and by 15mm it is big enough for a full frame 35mm sensor.

However lenses such as the EF-S 17-55 f/2.8 vignette at all focal lengths on full frame. I believe the constant max aperture has something to do with it, as otherwise the total amount of light coming out of the lens would somehow change despite the iris and entrance pupil remaining the same size.

Note that the reverse is not necessarily true, the image circle doesn't necessarily get larger with a variable max aperture lens. Some Nikon lenses contain a rear baffle to reduce flare, and this causes the same vignetting regardless of focal length. Integral petal hoods on ultra-wides can have the same effect.

Finally, assuming the DX 16-85mm does not vignette on full frame at 24mm, it will have the same angle of view as any other 24mm lens - 73.7 degrees horizontally and 53.1 degree vertically.

Conditional cropping based on focal length would produce a very strange relationship between focal length and angle of view with the lens getting wider then narrower as you zoom in!