Goats would be happy with an open shelter, they will just shelter if it rains heavy, then off out again, light rain they seem happy with.
I don't think just cutting docks would kill them, it would take many cuts to weaken them, digging them would be your best option, a small area at a time, making sure the others don't seed.
My thoughts when digging hundreds (maybe thousands) on thistle is 'divide and conquer ', small areas so I don't feel overwhelmed.