The main reason is simple: a shaved crop-sensor fisheye on full frame gives a unique, wider view than a full-frame diagonal fisheye, without the cost of buying another lens.

From the answers, the benefits are:

• wider field of view than Nikon’s 16mm full-frame fisheye
• a distinctive look that is different from a standard full-frame fisheye
• lower cost if you already own the crop fisheye
• useful coverage for 360x180 panoramic work, where wider coverage can reduce the number of shots needed

Why not just buy a circular fisheye? Circular fisheyes can have tradeoffs: compared with diagonal fisheyes, they may show more distortion, lower sharpness, and more chromatic aberration. They can also cost more.

So shaving the hood is a compromise: you permanently modify the lens and lose the warranty, but you gain a very wide fisheye view on full frame that may be cheaper and better suited to some pano workflows than buying a dedicated circular fisheye.

In short, people do it for field of view, creative effect, pano efficiency, and savings—not because it’s the “proper” solution.