Anything that has permitting factors - specifically, in this case, for the consumption of the consecrated offerings - will, depending, avoid the complicating factor of pigul. If pigul is established with regard to the peace-offering (shelamim), how can it apply to other offerings that aren't quite parallel to the shelamim? The Gemara explains it all very clearly, and pigul indeed is extended even to the oil that the person recovering from tzara'at offers (including non-shelamim meat offerings, bird offerings, and grain offerings). Such a clear delineation of pigul! Also, how it is that kohanim can eat from bird offerings. Plus, the grain offerings and atonement coming from the less likely source of the grain and the birds. With verses to teach it all.