I would use weed killer, I painted a lot leaves over the last couple of years and it does work. I've still got them but no where near as bad. I dig them out now but they do have strong tap roots and is difficult to remove completley. I'd cut the flowers off as well.
Saw the buttercup bit of Springwatch, what a difference between the two. The meadow buttercups look to me the type that you hold under your chin