I've encountered generally four different mating methods:
1) a brass ferrule is crimped on the stripped wire and then the ferrule is crimped to the barrel of the alligator clip
2) the stripped wire is wrapped around a mounting screw on the underside of the alligator clip
3) the stripped wire is soldered to the clip inside the little opening on the underside near the screw hole
4) the stripped wire is bent parallel to the jacket and that whole length is crimped to the barrel of the alligator clip (which is what you have)
Pomona and Mueller Electronics used method 1, but they have both gone down in quality immensely over the years.
Most older clips used method 2.
Anyways, if you want something done "right" you have to do it yourself. It's actually pretty easy to fix these clips: Just pull the wire out and then restrip and solder the wire to the clip using the slot on the underside. I also crimp the jacket to the barrel and add a spacer if the jacket is a little too small to crimp properly. This prevents rotation of the wire inside the barrel.
The brass ferrules are low resistance, but fail if the jacket is too short to fit properly inside the barrel--causing the strands to snap off near the ferrule. The soldering method works fine though, as the jacket has plenty of room in the barrel to prevent the strands from flexing too much and the soldered strands are far enough away from the part that flexes. The screw method is similar and works well, but adds bulk to the clip and doesn't look as nice.
Whoever complained about soldering melting the wire jacket has never made a clip using that method. Yes, a tiny and insignificant amount of jacket may melt, but the part that is near the end of the barrel is fine and so it doesn't matter. If you pre-tin the wire and use good flux with a high heat soldering iron, the solder joint can be made fast enough that the jacket near the joint is negligibly impacted.