Well there are quite a few issues to solve or complicate food if you can eat entire aspects of a creature
1) Dual Purpose materials: Do you want to EAT the chicken skin or craft with it?
2) Rotting
3) SIZE!: Goodness what is with this cat feeding the whole fortress? Small Livers should mean small meals
-Size should make cooking more important as a method to string together small foods that would barely feed a single dwarf and turn them into large filling dishes. Sausages!
4) Inedible parts: A Polarbear's liver is so Vitamin A drenched that it is downright poisonous
5) Use for Butchery Skill?: Perhaps the level of butchery should determine how much meat you get from an animal
6) Animals eat: Yes this helps too!
7) Cooking is less efficiant: Cooking often seems to multiply your in stock food... Is that how it works in real life?
Eating when not hungry: Dwarves should be able to eat even when they are not hungry such as during parties or because they are simply have a love of food.
9) Travel on their stomach: Needing large supplies of food to move dwarves across the map is a great way to go around it
10) Sustain Yar Settlements: New settlements to survive, whome have not had the time to make sufficiant farms, could require sizable food donations.
11) Blight, drought, and other food absorbers: Sometimes you simply cannot find food
12) Disease: Specifically that the food harms those who eat it... Diseases that cause food to deplete goes with the above
Some things I think shouldn't be an issue is however
1) extinction: Except in places with very sensative ecosystems... a small fortress shouldn't manage to extinct the forest... Even then why shouldn't we be able to send hunters outside the boarders of our fortress? I mean Civilisation sites do as it is anyhow. This especially applies to Fishing...
2) No suitable farming anymore: There should be options for dwarves who unfortunately cause their soil to become toxic... Even if that means sending people outside of the available area to find rich soil
These arn't suggestions (except the last two things) just possibilities for ways that food doesn't become overabundant even with the new organs as food.
Actually I was looking over the raws and apperantly there is an option for a tamed animal to be eaten as it is... Though I think it currently only applies to Vermin I think technically in the future it could apply to other animals... Giant Chocolate bunnies is becoming a real possibility.