From what I've seen, spoken to owners, and read about bedlingtons, yep, rabbiting and alround/general purpose dogs, including taking feathered game, is how they've been described.
My ex brother in law owned one which was mad on mink, as well as catching rabbits and foxes pretty regularly in cover. Superb nose on it, but was pretty dog aggressive.
He also owned two bedlington lurchers, at separate times, which also had great noses.
He has a whippet now, that is, in his words, far easier to live with and easier to control.