In article , Emrys Davies
writes

Forget your weed killer. Brambles are shallow rooters and if you cut
them down to a level where you can operate without being scratched you
can prise the roots with a fork and drag out the offending remnants
quite easily. A stout pair of gloves will be a great help.

I agree that no weed-killers are necessary. In fact there's no real need
to take out bramble root systems unless the space is wanted for planting
etc. If top re-growth is removed by cutting off stems when they are
large enough to see and handle, the roots below will begin to
deteriorate. After two or three removals of top re-growth even large
clumps of bramble roots will begin to die off in the ground.
--
Alan & Joan Gould - North Lincs.