Early DX cameras still had FX size mirrors, so there was no possibility of making DX lenses with a shorter back-focus, which would also allow the lens to be more compact. I suppose the long back focus was also retained to ensure the lens was more telecentric - digital sensors perform best if the ray angle is close to perpendicular. However I think modern DX now use smaller size reflex mirrors, and DX lenses with enough back-focus to clear a DX-size mirror would be no worse than an FX lens on an FX camera.
DX lenses with a shorter back-focus would of course be incompatible with FX cameras - when the picture is taken the reflex mirror would hit the rear of the lens. Users could be prevented from mounting DX lenses by modifying the mount so they wouldn't fit (in a similar way TCs have an extra tab so they only fit big telephoto lenses). Maybe Nikon thought it would be useful if DX lenses could mount FX cameras in DX mode. I'm not sure how useful this feature is, I suspect most owners of FX cameras rarely use DX lenses.
If DX lenses could be made with a shorter back-focus and the aperture was dropped from f/2 to f/2.8, a DX 16mm lens could easily be made much more compact. Even without a shorter back-focus, it should be possible to make DX primes which are relatively compact if the maximum aperture is not too fast. For example, the DX 16-80/2.8-4 already has a smaller filter size (72mm) than the Samyang 16/2 (77mm), a prime 16/2.8 would be smaller still.