Actually Rod,I'm surprised that you said you were able to solder the wire back on.
Rheostat wire is made of Nichrome which is how it attains it's resistance.It is also a hard steel and solder will not take to it which is why it is riveted.I have tried to solder Nichrome when I was a young apprentice almost 20 years ago and the boss looked at me like I was an idiot then told me the story.Maybe the solder is sticking to the rivets you are using.In some cases solder will completely surround a metal and look stuck?I'm always learning though so I could be wrong.
Cheers,Jamie.