Picture trying to machine a rivet's narrow end in a lathe. How would you do it? The jaws cannot solidly grab the rounded end. Even if they did, the rest of the rivet's body would be unsupported. With a small side force, the rivet tips off center.
Here is how I deal with turning the small end of a rivet-shaped object.
Your comments are welcome. All of us are smarter than any one of us.