This story is inaccurate. C14 naturally occurs and every living thing has C14 in it. When the animal dies, the C14 is not replaced and decays. That is the basis of Carbon dating.
(Unlike in archaeology studies, the half-life of 14C, about 5,700 years, does not affect these results.)
IMHO, the study took advantage of the increase of carbon 14 during that period of atmospheric bomb testing compared to the normal amount of carbon 14 in nature.