I'm not being hyperbolic, parabolic, elliptic, or any other form of conic section, degenerate or not. As far as I can tell, every part of your post was incorrect. It's a matter of honesty, if you've learned this incorrect information from somewhere, point me at it so I can give them a piece of my mind, CH.
Note that there's more to eating it that can pass the disease. Preparing it can easily lead to disease via contact with the body fluids, esp. for diseases like rabies. In the lack of waterproof latex gloves, the proper procedure is to use mage hand or unseen servant to do the work. As to the one-day spoilage, the main danger comes from toxins produced by anaerobic bacteria; specifically, I mean the gut flora. Has the intestines exploded yet from gas buildup? If it has not, then a cooking check to dress the carcass correctly (gut it, skin it, etc) avoids it. If it has or the DC 10 Cooking check fails, I would inflict Filth Fever in severe cases of undercooking or just a DC 12 1d2 Con poison for food poisoning in light cases, since the microbes themselves are dead but the toxin's not denatured.