Dirt,
If you can get some Image, do a mix of Roundup and Image plus Dawn dishwashing detergent (a teaspoon in a gallon). The Image will get even blackberries and the bindweed is severely damaged. You are right; those roots go to CHINA.
Don't get this mix anywhere near your favorite plants!
I have one spot in one bed with a few plants of bindweed. Each year it gets less healthy looking but this is 6 years of watching and spraying. One portion of the hillside lawn has it so I decided to keep that in lawn rather than fight bindweed.
Kevin