If you're happy enough with the clinch (minus the extra "improved" step), I find it easy to do with forceps: Pinch the line and tag end between same thumb and finger, leaving a bit of tag end sticking up, and the fly loose down in the loop formed. Insert closed forceps into the loop. Twist the loop with the forceps, making sure the fly doesn't get up into the twists. Grab the tag end with the forceps (still in the loop) and pull through.

There is also a tool to do the equivalent, but I haven't tried it.