Round-up used to work on those vines mainly in late summer when it could get dragged into the system during fall.
Otherwise Crossbow or something like that is better during the entire year.
I just looked at the label of the new Round-up today. Noticed it has diquat in it. Forgot about what that stuff does.
Maybe it will change things.
When blackberry removal is not a crisis regarding time, I prefer to spray, let it die, let it sit for about 2/3s of a year, and then it shatters.
You could take a 2x2 or broomstick and hit it, and it will just shatter into little pieces that can be raked up.
If time is short, then I prefer to use gas hedge shears. It cuts the vines to shreds. Then after raking, finding the crowns of the little stumps for digging is fairly easy.