I don’t know if this will work on brambles, but for pesky root systems I’ve had luck with Bonide’s Stump and Vine killer. You cut near the base of one of them, then paint the exposed stem with this stuff. It absorbs into the root system and kills all of it. Works great on pokeweed.

Edit: Turns out this is just a specific brand of triclopyr herbicide like MoonMelon mentioned. So here’s another recommendation for triclopyr!