Without being rabid, I lean towards organic methods. ...but if some weed ticks me off, I would use a herbicide or a nuclear bomb.
We've tried a couple strengths of Round-Up on our driveway, which is well away from our well, our garden, and our horse pastures. For the cost, it doesn't work for squat. We've had better luck with an attachment that hooks up to a propane tank and flames the weeds.
Regarding buttercups: Here we have what are called Ranunculus repens. When Baby got laminitis, we had to make a dry lot quickly. Prior to that, when allowed to graze freely, ours never touched the buttercups. When we made the dry lot, we took weed-whackers and cut the buttercups short (kind of bounce it off the ground to get them as short as possible.) The girls dug down and ate every root they could find, probably including lots of buttercup roots. It didn't give me a warm, fuzzy feeling that they were eating potentially toxic buttercups/roots, but so far all is o.k.