What do you eventually want to grow there? All those roots are holding
the soil from washing away; you don't really want to pull them out, you
just think you do. You want to kill them, and replace them with some
deep-rooted grass or something to hold the soil when the bramble roots
eventually rot. Or else you need to terrace that area.
What exactly did you spray them with? Glyphosate (Round-up) might work,
but something with Triclopyr (like Garlon, or Ortho brush killer) would
be better. Glyphosate will also kill the grass, Triclopyr or 2,4-d will
not. If you use 2,4-d please just spot-treat with it and don't spray
the whole area.
There's an old product called "Ammate X" that works good on brush, and I
think it breaks down into nontoxic nitrogen compounds that are actually
good for the soil. I don't know if it's still available.
Bob