It worked pretty well, actually.
It was either this or I had no use of either hand as I held ice against the thumb.
My thumb is almost fully recovered now. It was touch-and-go a couple of days, but I was able to save the whole hand (name that movie!). I've accidently bumped the thumb a few times and that makes it hurt like the dickens all over again. It's still a bit bruised and I have a bump of calcium deposit at the point of impact, but it could be worse and overall I'm pleased with my body's ability to heal itself.