I do run my goats through brush followed by the pigs. It works well. I am aware of the copper issue, another reason to not keep them together, for me at least, although some do it and say it works fine.
I plan to wether male offspring of my goats and run them through the brush. I don't want my girls out in quite that much brush, and heaven knows what funny milk flavors we would end up with!
My land was unfortunately logged before we ended up here so there is everything from grassy sections to low weeds/brush to small trees. It's a mess, sadly. But all edible to one creature or another!