How large an area are you dealing with? The roots can be dug out, which is best done in February where I live, but I don't know about where you're located.
I'm not physically able to do heavy work like that anymore, so I hired a crew to do it for me. I got the name of the company from the Extension Service office in my city. It was a company that specialized in removing invasive plants from ecologically sensitive areas.
I had an area about 15' x 30' that was solid blackberry bramble, 7' high or more. The 3-man crew had it completely cleared within 4 hours. They used mattocks and hoedads to do the work. It wasn't cheap, but it wasn't astronomical, either, and IMO, it was well worth every cent I paid.
That was 4 or 5 years ago, and the area is still clear of blackberries. I've pulled maybe half a dozen tiny new plants since then, but they had very small root systems. I'm pretty sure they weren't suckers from the old root system, because I just pulled them up by hand with no problems.
One thing I do know about wild blackberries is that just cutting them off at the ground only encourages them to send up new shoots from the roots. Goats would probably work, but you would have to have them around all the time to keep the brush cleared back. A friend of mine rented some goats to clear the blackberries on her property. She discovered, much to her chagrin, that the goats first ate everything except
the blackberry bushes. Maybe they were saving them for dessert!
"Isn't it enough to see that a garden is beautiful without having to believe that there are fairies at the bottom of it too?" - Douglas Adams